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UNITED STATES OF AMERICA. 



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Digitized by the Internet Archive 
in 2011 with funding from 
The Library of Congress 



http://www.archive.org/details/culminationofsciOOsmit 



THE CULMINATION 



OF THE 



SCIENCE OF LOGIC 

WITH SYNOPSES OF 

ALL POSSIBLE VALID FORMS OF CATEGORICAL REASONING IN 
SYLLOGISMS OF BOTH THREE AND FOUR TERMS. 



Vf1| JOHN 0*-' SMITH, 



A MEMBER OF THE BROOKLYN BAB. 




PUBLISHED BY 

HERBERT C. SMITH, 
16 COURT STREET, BROOKLYN. N. Y. 






IrmB LIBRARY 
Off CONGRESS 

WASHINGTON 



Copyright, 1888, by Herbert C. Smith. 



Electrotyped by B.. Harmer Smith & Sons 82 Beelnnan St.. New Yorit. 



PREFACE 



The following are two chapters of a treatise now in 
course of preparation, and to be entitled "Logic as a 
Pure Science, illustrated only by means of symbols indefi- 
nite in material, but definite in logical signification, with 
synopses of all possible valid forms of categorical reason- 
ing in syllogisms of both three and four terms." 

The preparation of the treatise was undertaken with 
but little expectation that it, or any part of it, would 
ever be published ; and certainly, with no thought of its 
resulting in any new contribution to the science. 

The author had long thought an elementary treatise 
on Logic as a pure science, with matter wholly elimi- 
nated, a desideratum ; and if any such has ever appeared, 
he is not aware of it. He acknowledges, however, that 
his acquaintance with the literature of the science is 
very limited. In writing the treatise, therefore, no con- 
crete examples were employed, but only those with sym- 
bols indefinite as to matter, but made definite as to 
their logical signification. 

The symbols adopted were the letters N, D, and J, to 



IV PKEFACE. 

represent the Minor, Middle and Major terms of the Syl- 
logism ; they being the middle letters of these words 
respectively. S, M and P are usually employed, as the 
initials of Subject, Middle and Predicate, but S and P 
are objectionable, being equally applicable to the sub- 
ject and predicate of the premises (as propositions), in 
each of which but one occurs in the statement of Syllo- 
gisms, and that one in its appropriate place in such 
representation in both premises, only in Syllogisms in 
the first figure ; in one premise only, in the second and 
third figures ; and in neither, in the fourth ; and their 
dual possible representations tend to confusion. Dis- 
tribution and non-distribution are signified by the use 
of capitals to represent terms distributed, and small 
letters, terms not distributed. Negation, in universal 
propositions, is indicated by crossing the capital let- 
ter representing the subject. The copula is expressed 
hy the characters, " — " for "is," and " — ■" for 
"is not." 

In translating the symbols and characters as em- 
ployed in propositions into spoken language, the sig- 
nification of the symbols should of course be expressed 
in respect to the subject, but implied in respect to the 
predicate , according to common usage and the well- 
known rules that all universal propositions (and no par- 
ticular) distribute the subject, and all negative (and no 
affirmative) the predicate. 



PREFACE. V 

Thus the four propositions, A, E, I, O, when written 
in symbols and characters as above, should be read and 
understood as follows : 

(A) D — j All D is j {meaning All D is some J) 

(E) B-— J No D is J ( " No D is any J) 

(I) d — j Some D is j ( " Some D is some J) 

(0) d — -J Some D is not J ( " * Some D is not any J) 

The consideration of Hypotheticals was reached in 
the preparation of the treatise, and in the course thereof, 
analyses of conditional propositions of both three and 
four terms, in all forms in which they can be expressed, 
were made ; and the study of their results led to 
the gradual unfolding of the doctrine of Sorites con- 
tained in the second of the following chapters. 

That doctrine is the culmination of the Science of 
Logic, which without it has hitherto been incomplete. 

The treatise, up to this point, had been written 
wholly in short-hand, and to guard against the possi- 
bility that the discovery might be lost if the author 
should not live to finish it, and the notes should not be 
deciphered, these chapters were written out in full, and 
put in position where they would be found and pub- 
lished, in such contingency. 

But, inasmuch as the work yet remains to be com- 
pleted, and the notes to" be written out (which can only 
be done by the author, his system of short-hand being in 
many respects peculiar), its appearance will be consider- 



VI PREFACE. 

ably delayed ; and as the discovery, when made known, 
will, it is believed, not only be an occasion of interest 
from a scientific point of view, but will prove also to 
be of practical utility, the author has determined to 
publish these two chapters in advance. The chapter 
on Enthymemes is published as preliminary, and to ex- 
hibit the synopses therein contained (of which the last 
shows all valid simple Syllogisms [of three terms] at full 
length and in regular form), in connection with those 
contained in the chaj)ter on Sorites (Syllogisms of four 
terms), thus bringing together, as it were in one view, 
all possible valid forms of categorical reasoning, To 
those for whose benefit they are thus published the 
chapters may seem to be unnecessarily diffuse and 
minute, but to condense them would involve very con- 
siderable labor, and they are therefore put forth in the 
form in which they were written to take their appro- 
priate places in the full treatise, trusting that their 
minor defects and redundancies may be overlooked. 

If the remainder of the treatise shall never appear 
from the author's pen, there will be little or nothing 
lost. The suggestion herein made, if it have any merit, 
will lead other and abler pens to supply the desideratum. 

Brooklyn, January 14, 1888. 



OF ENTHYMEMES. 

§ 1. We have hitherto considered the process of 
reasoning with three terms, categorically, in its full 
expression, and have examined all the possible forms 
of such expression. Such forms are seldom resorted 
to, either in common conversation or formal discourse, 
whether spoken or written, but abridged forms of argu- 
ment are employed in which only part of the process 
is expressed, the remainder being implied, and being 
usually so obvious as not to require expression. We 
come now to consider such abridged forms. 

They are called Enthymemes. 

§ 2. An Enthymeme is a Syllogism of which but two 
propositions are expressed, the third being implied. 

Enthymemes are of three orders ; 

1st. That in which the major premise is implied. 
2d. That in which the minor premise is implied. 
3d. That in which the conclusion is implied. 



The following are examples. 


Of the first : 








N 


- d; 




\ X 


— J- 


Of the second : 








D 


— j; 




\ N 


— j- 



VI PREFACE. 

ably delayed ; and as the discovery, when made known, 
will, it is believed, not only be an occasion of interest 
from a scientific point of view, but will prove also to 
be of practical utility, the author has determined to 
publish these two chapters in advance. The chapter 
on Enthymemes is published as preliminary, and to ex- 
hibit the synopses therein contained (of which the last 
shows all valid simple Syllogisms [of three terms] at full 
length and in regular form), in connection with those 
contained in the chapter on Sorites (Syllogisms of four 
terms), thus bringing together, as it were in one view, 
all possible valid forms of categorical reasoning. To 
those for whose benefit they are thus published the 
chapters may seem to be unnecessarily diffuse and 
minute, but to condense them would involve very con- 
siderable labor, and they are therefore put forth in the 
form in which they were written to take their appro- 
priate places in the full treatise, trusting that their 
minor defects and redundancies may be overlooked. 

If the remainder of the treatise shall never appear 
from the author's pen, there will be little or nothing 
lost. The suggestion herein made, if it have any merit, 
will lead other and abler pens to supply the desideratum. 

Brooklyn, January 14, 1SSS. 



OF ENTHYMEMES. 

§ 1. We have hitherto considered the process of 
reasoning with three terms, categorically, in its full 
expression, and have examined all the possible forms 
of such expression. Such forms are seldom resorted 
to, either in common conversation or formal discourse, 
whether spoken or written, but abridged forms of argu- 
ment are employed in which only part of the process 
is expressed, the remainder being implied, and being 
usually so obvious as not to require expression. We 
come now to consider such abridged forms. 
They are called Enthymemes. 

§ 2. An Enthymeme is a Syllogism of which but two 
propositions are expressed, the third being implied. 
Enthymemes are of three orders ; 

1st. That in which the major premise is implied. 
2d. That in which the minor premise is implied. 
3d. That in which the conclusion is implied. 

The following are examples. 

Of the first : 





N - 


- d; 




\ K - 


- J- 


Of the second : 








D - 


- J; 



N — 



8 LOGIC AS A PUEE SCIENCE. 

Of the third : 

and N — d. 

In each case the three terms requisite to make up 
a full Syllogism appear, and the implied premise or con- 
clusion can be readily supplied. 

Enthymemes of the first order are herein called 
Minor, and those of the second order Major Enthy- 
memes, from the names of their expressed premises, 
respectively. 

§3. As every Enthymeme, together with its implied 
premise or conclusion, is a Syllogism, it is evident that 
only such can be valid as are symbolized by the letters 
by which the expressed propositions are symbolized, in 
the combinations of vowels symbolizing the propositions 
of all allowable moods of categorical syllogisms, as 
hereinbefore shown. 

By reference thereto, it will be found that all valid 
Enthymemes must consist of propositions of which the 
following are the symbols ; namely, 



Of the first order. 
(Minor Enthymemes.) 


Of the second order. 
(Major Enthymemes.) 


Of the third or 


— , A, A; 




A, -A 




A, A,-; 


- A, E; 




A, -, E 




A, E,-; 


-,A, I; 




A,- I, 




A, I, — ; 


-A, 0; 




A, - 




A, 0,-; 


— , E, E; 




E, - E 




E, A, -; 


- E, 0; 




E, - 




E, I, — ; 


- I, I; 




I, -, I; 




I, A,-; 


- I; ; 








- 0, 0. 




o, -, 0. 




0, A, -. 



ENTHYMEMES. 9 

The symbols of minor and major Enthymemes are the 
same, except that there is no valid major Enthymeme in 
I, .0. There are no valid minors in E, O, except useless 
ones. Leaving the latter out of consideration, it will "be 
found that A occurs four times as the symbol of the pre- 
mise, and but once as the symbol of the conclusion in 
both minor and major Enthymemes ; E once in minors 
and twice in majors as the symbol of the premise, and 
twice in each as the symbol of the conclusion ; I twice in 
minors and once in majors as the symbol of the j>remise 
and twice in each as the symbol of the conclusion ; and 
O once as the symbol of the premise and three times as 
the symbol of the conclusion in both minors and majors. 

Minor Enthymemes are the most common, the sup- 
pressed major premise being usually a general rule, 
readily recognized and acquiesced in without being 
expressed. 

Enthymemes of the third order are seldom employed, 
except in combination with one of the first or second 
order. They will be referred to when we come to the con- 
sideration of Sorites, and it will be found that they occur 
sometimes in the order of the symbols above shown, 
namely, major premise first, and minor second ; and some- 
times in the reverse order, minor first, and major second. 

§ 4. To the three orders may be added a fourth ; viz., 
an Enthymeme with but one expressed and two implied 
propositions. Every demonstrable categorical proposi- 
tion, put forth independently as the expression of a 
judgment, is such an Enthymeme, being the conclusion 
of two implied premises. If the question is asked, 
"What is ~N1" the answer must be either a random 



10 LOGIC AS A PURE SCIENCE. 

expression in the form of a proposition, but meaningless, 
or the result of thought more or less deliberate, and 
therefore based upon some reason, which, as we have 
before seen, is a just (or assumed as just) ground of con- 
clusion. This ground must be a mental comparison of 
the subject, 1ST, with some other term, and of that again 
with the term predicated of the subject in the answer. 
The premises thus formed, but not expressed, must be 
obvious to the questioner, when the answer is given, and 
therefore admitted ; or otherwise explanation would be 
demanded. Were this not so, there could be no reason- 
ing without going back in every process to some inde- 
monstrable proposition {axiom or postulate), or to the 
Great First Cause ; with which or with Whom, when 
reached in the process of investigation, we must necessa- 
rily set out in retracing our steps in the deductive pro- 
cess of reasoning. 

Such an Enthymeme may also consist, in so far as it is 
expressed, of a single proposition put forth as a premise 
(usually the major), the unexpressed premise and conclu- 
sion being left to be gathered from the attending circum- 
stances or from the subject-matter under consideration. 

§ 5. The middle term will of course be that term of 
the expressed premise, in minor and major Enthy- 
memes, which is not common to both propositions, and 
in Enthymemes of the third order, that which is common 
to both ; and will vary in position according to the fig- 
ure, and the character of the premise, whether minor or 
major. In minor and major Enthymemes it may or may 
not be distributed, according to the mood, and character of 
the premise, whether minor or major ; but in Enthymemes 
of the third order must be at least once distributed. 



ENTHYMEMES. 11 

§ 6. It is manifest, that there are three, and can be but 
three, Enthymemes having two expressed propositions, 
viz., one minor, one major, and one of the third order, 
in each allowable mood of the syllogism ; and as the 
number of such moods is twenty-four, including the use- 
less ones, so the number of Enthymemes of each kind is 
limited to twenty-four. 

The following are synopses of all possible valid forms 
of categorical Enthymemes of two expressed propositions, 
together with the implied premise or conclusion of each, 
as the case may be. On the first page of each of the two 
synopses of minor and major Enthymemes the forms of 
the expressed propositions are printed in full, each but 
once, in the order A, I, E, O, of the symbols of the con- 
clusion, but on the second page they are printed in full 
throughout. Where they are repeated, they will be 
found to have in each case a different implied proposi- 
tion. By counting, it will be found that there are fifteen 
forms of the expressed propositions of minor Enthy- 
memes (of which four are useless) and twelve of majors. 
The capital letters in the names of the moods on each 
page of the synopses are the symbols of the proposition 
or propositions in the column next adjoining. 

The synopsis of Enthymemes of the third order, will 
serve also as a synopsis of those of the fourth order, as 
first described, by considering the words "expressed" 
and "implied" as transposed in the headings over the 
columns of the propositions. 

As arranged on page 17, and read across the page, it 
exhibits all possible valid forms of categorical reasoning 
with three terms, at full length and in regular form, in 
the order of the Moods of the Syllogism. 



12 



LOGIC AS A PUKE SCIENCE. 



Synopsis of all Possible Valid Forms of -Categorical Enthymemes 
of the First Order. (Minors.) 

IN THE ORDER A, I, E, 0, OF THE SYMBOLS OF THE CONCLUSIONS. 



fe a £ 

o g | 










s « s 

o w 5? 


Expressed Propositions. 


Implied Proposition. 


Moods op 


g g o 

B M & 


Minor Premise. 


Conclusion. 


Major Premise. 


Syllogisms. 


A, A. 


N - d ; .-. 


■N - .]• 


v D-j. 


bArbara. 


A, I. 


N - d ; .-. 


11 - J- 


v D-j. 


A, a, i. 1st fig. 




D - n ; .-. 


n - 3- 


v D-j. 


dArapti. 








■•• d-j. 

•.• J - d. 


dlsamis. 
brAmantip. 




II 


a 


••• j - a. 


dlmaris. 


I, I. 


n — d ; 


n - ]'• 


v D-j. 


dArii. 




d — n ; 


n - J- 


v D-j. 


dAlisi. 


A, E. 


N - d ; .■ 


N — J. 


v D-J. 


cElarent. 




// 


// 


v J - D. 


cEsare. 


E, E. 


5* - D ; .-. 


N — J. 


v J - d. 


cAmestres. 




D-N; .■ 


ff — J. 


v J - d. 


cAmenes. 


A., 0. 


N - d ; .- 


ii -_ J. 


v D-J. 
v. J - D. 


E. a, o. 1st fig. 
E, a, o. 2d fig. 




D — n ; .• 


n -~ J. 


v D-J. 

v d -^ J. 


fElapion. 
b Okardo. 




n 


;/ 


v J - D. 


fEsapo. 


E, 0. 


SF-D; .• 


n ->_ J. 


v J - d. 


A,e,o. 2d fig. 




B- — N; .• 


n — J. 


v J - d. 


A, e, o. 4th fig. 


I, 0. 


n — d ; .* 


n-^J. 


v D - J. 


fErio. 




a 


// 


v J - D. 


fEstino. 




d — n ; 


n — J. 


v D-J. 


fEriso. 




a 


a 


v J - D. 


frEslson. 


0, 0. 


n -^ D ; .- 


. n — J. 


v J - d. 


bAroko. 



ENTHYMEMES. 



13 



Synopsis of all Possible Yalid Forms of Categorical Enthymemes 
of the First Order. (Minors,) 

IN THE ORDER OF THE MOODS OF CATEGORICAL SYLLOGISMS. 



Moods or 


Expressed Propositions. 


Implied Proposition. 


Syllogisms. 


Minor Premise. Conclusion. 


Major Premise. 


barb Ay A. 


N — d ; 


•• N - j. 


v D - j. 


celArEnt. 


N - cl ; 


N - J. 


v B- - J. 


«, A. I. 
darll. 


N" — d ; 
n — d ; 


n — 1- ) 


v D - j. 


e, A, 0. 
ferlO. 


N- d ; 
n - d ; 




v B - J. 


cesArE. 


N— d ; 


N - J. 


v J - B. 


camEstrEs. 


N-D ; 


N - J. 


J - d. 


e, A, 0. 
festin 0. 


TS — d ; 
n — d ; 




v. J - B. 


a, E, 0. 
bar Ok 


n -^ D ; 


n -~ J. ) 
n-wJ. j 


v J - d. 


darAptl. 


D — n ; 


n - j. 


••• D - j. 


disAmls. 


D-n; 


n - j. 


••• d - j. 


datlsl. 


d - n ; 


n - j. 


v D - j. 


f el Apt On. 


D - n ; 


n -~ J. 


v B - J. 


bokArdO. 


D - n ; 


n-^J. 


d-^J. 


ferlsO. 


d — n ; 


n^J. 


•■• B - J. 


bramAntlp. 


D - n ; 


n - j. 


•/ J - d. 


camEnEs. 
a, E, 0. 


B--N; 
B--N; 


N - J. ) 
ii-^-J. j 


v J - d. 


dimArls. 


D — n ; 


n - j. 


■•■ j - d. 


fesApO. 


D - n^ 


n -~ J. 


v J - D. 


f reels On. 


d - n ; 


n-~J. 


v J - D. 



14 



LOGIC AS A PUKE SCIENCE. 



Synopsis of all Possible Valid Forms of Categorical Enthymemes 
of the Second Order. (Majors.) 

IN THE ORDER A, I, E, 0, OF THE SYMBOLS OF THE CONCLUSIONS. 



& fi fc 










°3 m 5 

© B ! 


Expressed Propositions. 


Implied Proposition. 


Moods of 


s (2 d 

a 8 ft 


Major Premise. Conclusion. 


Minor Premise. 


Syllogisms. 


A, A. 


D-J 


••• N-j. 


v N - d. 


barb At a. 


A, I. 


D-j; 


•■■ n - ]'■ 


••• N" - d. 


a, A, i. 1st fig. 




// 


// 


v n — d. 


darli. 




// 


;/ 


v D - n. 


darApti. 




/; 


?/ 


v d - n. 


datlsi. 




J-d; 


••• a - i- 


v D - n. 


bramAntip. 


I, I- 


d - J ; 


•'• n - j. 


v D — n. 


disAmis. 




j-d; 


••• n - .1- 


v D — n. 


dimAris. 


A, E. 


J-d; 


.-. N - J. 


v N-D. 


camEstres. 




// 


// 


v B- — N. 


camEnes. 


E, E. 


D- J 


; .-. N-J. 


•.• N - d. 


celArent. 




J-D 


; .'. N-J. 


••• N - d. 


cesAre. 


A, 0. 


J-d 


; .-. n-v_J. 


v N-D. 


a, E, o. 2d fig. 




// 


// 


v n -~ D. 


bar Oko. 




// 


n 


v D- - N". 


a, E, o. 4th fig.. 


E, 0. 


fr-J 


; .*. n -~. J. 


v N — d. 


<?, A, o. 1st fig. 




a 


»; 


•.• n — d. 


. ferlo. 




n 


/'/ 


v D - n. 


felApton. 




it 


II 


v d — n. 


ferlso. 




J-D 


; .*. n -w J. 


v N" — d. 


e, A, o. 2d fig. 




// 


;/ 


v n — d. 


festlno. 




;/ 


// 


v D — n. 


fesApo. 




// 


// 


v d — n. 


freslson. 


0, 0. 


d^J; 


.'. 11 -w J. 


• v D — n. 


boJcArdo. 



ENTHYMEMES. 



15 



Synopsis of all Possible Valid Forms of Categorical Enthymemes 
of the Second Order. (Majors,) 

IN THE ORDER OF THE MOODS OF CATEGORICAL SYLLOGISMS. 



Moods of 


Expressed Propositions. 


IMPLIED PROPOSITION. 


Syllogisms. 


Major Premise. Conclusion. 


Minor Premise. 


bArharA. 


D- j 


; ••• N" - j. 


v N - d. 


cElarEnt. 


B-- J 


; .-. N - J. 


v N - d. 


A, a, I. 


D- J 


; ••• n - j- 


v N - d. 


dAril. 


D- j 


; .-. n - j. 


n — d. 


E, a, 0. 


B-- J 


; .-. n -~ J. 


v N" - d. 


fEriO. 


B-- J 


; .-. n -^ J. 


n — d. 


cEsarE. 


J -D 


; .-. if - J. 


v tf — d. 


cAmestrEs. 


J - d 


; .-. N - J. 


v ' ff-D. 


E, a, 0. 


J -D 


n -^ J. 


v N - d. 


fEstlnO. 


J -D 


11 -v_ J. 


. ;.• n - d. 


A, e, 0. 


J - d 


n -^ J. 


v Jf — D. 


bArokO. 


J - d 


n -w J. 


n -^ D. 


dAraptl. 


D- j. 


n - j. 


v D — li. 


dlsamls. 


a - i : 


n - j. 


v D - n. 


dAtisI. 


d- j 


n - j. 


d — n. 


fElaptOn. 


©•- J 


n -~ J. 


v D — n. 


bOkardO. 


d^J, 


n -^ J. 


v D - n. 


/ErisO. 


fr- J, 


n -^ J. 


d — n. 


IrAmantlp. 


J - d ; 


n - j. 


v D - n. 


cAmenEs. 


J - d ; 


if - J. 


v B- — N. 


A, e, 0. 


J - d ; 


n -w J. 


v B- — N". . 


dlmarls. 


.] - d : 


n - j. 


v D - ii. 


fEsapO. 


dF -D: 


- .-. n-wj. 


v D - ii. 


frEsisOn. 


£-D: 


n-^J. 


d — n. 



16 



LOGIC AS A PUKE SCIENCE. 



Synopsis of all Possible Valid Forms of Categorical Enthymemes 
of the Third Order. 

IN THE ORDER A, I, E, 0, OF THE SYMBOLS OF THE CONCLUSIONS. 



& « °° 








1 § 1 


Expressed Propositions. 


Implied Proposition. 


Moods of 


S & § 


Major Premise. Minor Premise. 


Conclusion. 


Stxxogisms. 


A. A. 


I) — j, and !N — d. 


•• N-j. 


barbarA. ' 


A, A. 


D - j, « N - d. 


■*■ n - j- 


a, a, I. 1st fig. 


A, I. 


D — j, " n — d. 


r/ 


daril. 


A, A. 


D - j, « D - n. 


// 


daraptl. 


A, I. 


B - j, » d - n. 


.*. " 


datisl. 


I, A. 


d — j, n D - 11. 


/' 


disamls. 


A. A. 


j _ d, " D - n. 


a 


bramantlp. 


L A. 


j — d. " D — n. 


.'. a 


dimarls. 


B, A. 


B- - J, " IS-— d. 


.-. X - J. 


celarEnt. 


E, A. 


j _ b, " X - d. 


.*. a 


cesarE. 


A, E. 


j _ a, „ jf - D. 


.*. " 


camestrEs. 


A, E. 


j _ d, " B - N. 


" 


camenEs. 


E, A. 


B- — J, » N - d. 


.-. n -^ J. 


e, a, 0. 1st fig. 


E, I. 


B- — J, " ii — d. 


.*. " 


feriO. 


E, A. 


B - J, " D - n. 


.*. " 


felaplOn. 


E, I. 


B - J, M d - 11. 


" 


ferisO. 


0, A. 


d -v_ J, " D — n. 


a 


bokardO. 


A, E. 


j _ a, " X - D. 


a 


a, e. 0. 2d fig. 


A, 0. 


J — d, " n -^ D. 


a 


barokO. 


A. E. 


j _ a, " B — N". 


a 


a, e, 0. 4th fig. 


E, A. 


£ - D, " K" - d. 


a 


e, a, 0. 2d fig. 


E, I. 


J _ b, " n - d. 


n 


festin 0. 


E, A. 


J — D, " B — n. 


a 


fesapO. 


E, I. 


3- — B, " d — n. ' 


a 


f rests On. 



.ENTHYMEMES. 



17 



Synopsis of all Possible Valid Forms of Categorical Enthymemes 
of the Third Order. 

IN THE ORDER OF THE MOODS OF CATEGORICAL SYLLOGISMS. 



Moods of 




Expressed Propositions. 


Implied Proposition. 


Syllogisms. 


Major 


Premise. 




Minor Premise. 


Conclnsiou. 


bArbAra. 


D 


- i> 


and 


X - d. 


■•■■ N-j. 


cElArent. 


B 


-J, 


r/ 


X - d. 


.-. X - J. 


A, A, i. 


D 


- i, 


ii 


X - d. 


••• n - j. 


dArli. 


D 


- j- 


n 


n - d. 


■•■ n - j. 


E, A, o. 


& 


-J, 


" 


X- d. 


.-. H-vJ; 


fErlo. 


D- 


-J, 


n 


n - d. 


.-. n-vJ, 


cEsAre. 


J 


-D, 


n 


X - d. 


.-. X-J. 


cAmEstres. 


J 


- d,. 


ii 


X-D. 


-. X - J. 


E, A, o. 


J 


- D, 


ii 


X- d. 


.-. 11 -~ J. 


fEstlno. 


J 


- D, 


V 


n — d. 


11 -w J. 


A, E, o. 


J 


-d, 


ii 


X- D. 


.-. n-^J. 


bArOko. 


J 


- d, 


n 


n-_D. 


.-. n--J. 


dArApti. 


D 


- j, 


ir 


D - n. 


••• ii - j- 


dlsAmis. 


d 


- h 


u 


D - n. 


••• n — j. 


dAtlsi. 


D 


- h 


n 


d — n. 


••• n - j. 


fElApton. 


B- 


-J, 


ii 


D - n. 


.-. n-_J. 


bOkArdo. 


d 


— J, 


ii 


D - n. 


.-. n -~ J. 


fEr/so. 


B- 


-J, 


ii 


d — n. 


.-. n-~J. 


brAmAntip. 


J 


-d. 


ii 


D - n. 


••• n — j. 


cAmEnes. 


J 


-d, 


ii 


B--X. 


.-. X - J. 


A, E, o. 


J 


-d, 


n 


B-— X. 


.-. ii-- J. 


dlmAris. 


J 


-a, 


ii 


D - n. 


•'• ii - j- 


fEsApo. 


J 


-d, 


ii 


D - n. 


.-. n -~ J. 


frEsIson. 


J 


-D, 


ii 


d — n. 


.-. n--J. 



18 LOGIC AS A PURE SCIENCE. 

§ 7. The following will serve as rules by which the 
implied proposition of every Enthymeme having two 
expressed propositions may be supplied, the first ,being 
applicable to those of either the first or second order, 
and the second to those of the third. 

1st. The term of the conclusion of an Enthymeme 
of either the first or second order which is 
common to both expressed propositions deter- 
mines the character of the expressed premise, 
whether minor or major, according as the 
same is either the subject {minor term) or 
predicate {major term) of the conclusion, and 
the implied premise may be found by com- 
paring the other two terms. 
2d. The term of the expressed minor premise of 
an Enthymeme of the third order not common 
to both premises is the subject, and that of 
the expressed major premise not common to 
both is the predicate, of the implied conclu- 
sion, which is universal or particular, and 
affirmative or negative, as called for by the 
premises. 



OF SORITES. 

§ 1. We come now to the consideration of reasoning 
with four terms, categorically ; and we shall herein- 
after find that that is the limit beyond which the 
human mind cannot go. 

§ 2. If we set out to make an investigation concern- 
ing any subject, N, and, in the process of our investiga- 
tion, become possessed of three judgments, which we 
put forth in the form of propositions, as follows : 

N — d; 

D - j; 

J -x, 

we may at once apply to such propositions the dictum 
of Aristotle, by extending it, as follows — 

I first repeat the dictum : 

"Whatever is predicated (i. <?., affirmed or denied) 
universally, of any class of things, may be predicated, in 
like manner (viz., affirmed or denied), of any thing com- 
prehended in that class." 

As extended it will read : 

Whatever is predicated (i. e., affirmed or denied) uni- 
versally, of any class of things, may be predicated, in 
like manner (viz., affirmed or denied), of any class com- 
prehended in that class; and, in like manner, of any 
thing comprehended in any class so comprehended. 



20 LOGIC AS A PUKE SCIENCE. 

We have in our last proposition predicated X (x) of 
the whole class J, and in the second proposition have 
shown that the class D is comprehended in the class J. 
X (x) may therefore be predicated of the class D. But 
we have also shown in the first proposition that N (which 
may be either a class, or some single thing) is compre- 
hended in the class D. We are therefore warranted, by 
the dictum as extended, in predicating X (x) of N ; viz. : 

N — x. 

Stating the propositions in their reverse order, and 
appending to them the proposition thus justified, with 
the word ' ' therefore ' ' x^refixed, we shall have the follow- 
ing expression, which is a Sorites ; viz. : 

J - x; 

D- j; 

N — d; 
.-. M" — x. 

But we may, without reversing the order of the 
propositions, append the new proposition, and will have 
the same Sorites, biit in a different form ; viz. : 

N — d; 

D- U 

J — x; 

.-. N — x. 

The conclusiveness of the reasoning in both forms is 
apparent. 

§ 3. Thus we have a complete Syllogism (but in two 
different forms or figures) consisting of four propositions, 
composed of four terms. 



SORITES. 21 

Let us now proceed to analyze it, and in the course 
of the analysis I shall give new names to the terms and 
propositions, which will be used when referring to them 
as parts of the Sorites, so as to distinguish them from 
like parts of a simple Syllogism, which will be called, 
when referred to as such, by their old names. 

And 1st, as to the terms. 

The subject, N, with which we set out, is equiva- 
lent to the minor term as we have hitherto employed 
it. I call it the magnus term of the Sorites, in the 
sense of holding a chief position ; it being the principal 
thing about which we are concerned. 

The two terms, D and J, are each greater {major) 
than the magnus term in the forms above exhibited 
(which you will hereinafter find are the perfect forms), 
but one, D, is less {minor) than the other, J. They are 
both middle terms, and are each once distributed, and 
are compared, one with one of the other terms, and the 
other with the other, in the first and third propositions, 
and with each other in the second. They will be called, 
D, the minor-middle, and J, the major-middle terms. 

The term X is equivalent to the major term as 
hitherto employed, but is greater than the major-middle 
term, and is the greatest of all the terms of the Sorites. 
It will therefore be called the maximus term. 

The four terms, as in the case of a simple Syllogism, 
occur twice each, the ^magnus and maximus terms each 
once in the premises (first three propositions) and once in 
the concluding proposition, and the minor-middle and 
major-middle terms each twice in the premises. 



22 LOGIC AS A PURE SCIENCE. 

N and X are letters in the words magnus and maxi- 
mus respectively, and will serve to keep their logical 
significations in mind, in like manner as the letters ~N, 
D, and J, in the words minor, middle, and major, have 
hitherto served in respect to their logical significations : 
but they will not in their future use so serve invari- 
ably. 

2d. As to the propositions. 

Three are premises, and will be called from the names 
of the terms occurring in them respectively : 

The magnus premise ; 

The middle premise (omitting the prefixes minor and 
major as unnecessary, there being no middle premise 
in a simple Syllogism) ; 

The maximus premise. 

The concluding proposition will hereinafter be found 
to be the ultimate one of two conclusions warranted by 
the premises ; and to distinguish it as such, I shall call 
it the ultima (conclusio understood). 

3d. As to the figure. 

The figure of a simple Syllogism depends upon the 
positions of its terms, but that of a Sorites upon the 
positions of its magnus and maximus premises. It 
will be called the configuration. There are two, the first 
called regressive, in which the maximus premise is the 
first, and the magnus last ; and the second, %>^ogresswe, 
in which the magnus premise is the first and the maxi- 
mus last. The progressive configuration was the only 
one known until about the beginning of the seventeenth 



SORITES. 23 

century, when the regressive was discovered by a Ger- 
man logician named Groclenius ; and it is called also 
Goclenian after him. It has been a subject of dispute 
among logicians as to which configuration should be 
called progressive, and which regressive, but the prevail- 
ing opinion is in favor of the names as herein used. 
They are generally treated of in the order as in the last 
sentence ; but I have reversed it, exhibiting the regress- 
ive first, and the progressive last. The moods of each 
configuration, and their number, will hereinafter appear. 

§ 4. If all Sorites, in respect to the positions of the 
terms,. were in the forms hereinbefore given, and their 
conclusiveness were equally as apparent, I might at once 
proceed further to illustrate and comment upon them, 
and state the rules usually given in logical treatises con- 
cerning them, which are applicable only in such case ; 
but such is not the case, and I defer further comment 
until I shall have exhibited them in another aspect in 
which they can be considered ; viz., as complex expres- 
sions consisting of two Enthymemes. 

The Sorites, so to be exhibited, will be the same as 
before given ; and for the sake of brevity, I shall call 
the terms and propositions by the names hereinbefore 
given to them, in advance of exhibiting them under the 
new aspect. 

§ 5. For the purpose of such consideration I repeat 
the three propositions with which we set out. 

N — d; 

d - j ; 

J — x. 



24 LOGIC AS A PUEE SCIENCE. 

If now, having possessed ourselves of these judgments, 
but failing to observe, from their perfect concatenation, 
that we may at once deduce from them the ultimate con- 
clusion wrapped up in them, we proceed to syllogize 
with them by means of simple Syllogisms of three prop- 
ositions, we shall naturally commence with the widest 
truth which we have discovered, viz., J — x ; and we 
shall find our first Syllogism to be as follows : 

J - x; 

D- j; 

.-. D - x, 

and, having thus become possessed of a new truth, viz., 
D — x, we shall put it forth as a premise, combining 
with it our first proposition, as yet unemployed, and 
produce a second Syllogism as follows : 

D -x; 

N — d; 
.-. N — x. 

The conclusion of this second Syllogism is the ultima 
of the Sorites, as we have before seen it. 

But if, in the course of our investigation, we had 
stopped after the discovery of the first two truths, viz. ; 

N — d; 
D - j, 

and had syllogized with them, we should in like manner 



SORITES. 25 

have commenced with the widest truth then discovered, 
viz., D — j, and our first Syllogism would have been : 

d — j ; 

N" — d ; 
••• N - j. 

The question would then naturally have arisen, But 
what is J % and resuming the process of investigation, 
we should have discovered that J — x, and thereupon 
would have syllogized again : 

J -x; 

n- j; 

.-. N — x, 

and thus, by a second series of Syllogisms, we should 
have arrived at the ultima of the Sorites, as we have 
before seen it. 

By the former process, we retraced our steps after 
having reached the summit of our investigation, and 
it is therefore properly called regressive ; by the latter 
we have reasoned as we progressed, and it is therefore 
properly called progressive ; but by both processes we 
have arrived at the same ultimate conclusion, illustrat- 
ing the aphorism that "all truth is one." 

The middle premise, as you will observe, is the minor 
premise of the first Syllogism in the first series, and the 
maximus premise the major ; and the middle premise 
is the major premise of the first Syllogism in the second 
series, and the magnus premise the minor ; and all the 



26 LOGIC AS A PURE SCIENCE. 

Syllogisms are in Barbara in the first figure, which yon 
have learned is the only perfect figure. 

§ 6. But Ave may reason imperfectly, and that too, 
even when we have our judgments in a perfect concate- 
nation, as they have thus far been exhibited ; and, in 
such case, we shall find our Syllogisms to be in one or 
more of the imperfect figures. If, in the regressive 
process we begin to syllogize with the middle premise 
as the major premise of the first Syllogism (instead of 
the minor), and the maximus as the minor (instead of 
the major) ; and in the progressive process, with the 
middle premise as the minor premise of the first Syllo- 
gism (instead of the major), and the magnus premise as 
the major (instead of the minor), we can frame, or 
attempt to frame, two other series of Syllogisms, which I 
here exhibit, with the two Syllogisms of each series, 
side by side, as follows : 

In the regressive process. 

D — j; ^ x — d; 

J — x; ¥ — d; 

.-. x — d. ^^ 

In the progressive process. 

N — d; J — x; 

D — j ; — j — n ■; 

.-. j — n. — .-. n — x. 

In the latter series, only a particular ultimate con- 
clusion is arrived at ; in the former, no ultimate conclu- 
sion is warranted by reason of non-distribution of the 
middle term in the second attempted Syllogism. 



SORITES. 27 

Thus, as yon will perceive, imperfect processes are 
followed by imperfect or no results. 

" § 7." To recur now to the two x^rincipal series, and for 
the purpose of bringing the two Syllogisms of each 
together, in such a method of arrangement that you may 
at once see the connection between them, and the appli- 
cation of the remarks that are to follow, I repeat them, 
putting the two Syllogisms of each, side by side. 





Fi 


•Sit, 


or 


regr 


essive 


series. 


J - 


- x; 










D - x; 


D - 


- i; 










N — d; 


. D - 


- x.— 










.-. N — x. 



Second, or progressive series. 

D-j; J-x; 

X — d ; ^ N — j ; 

... n — j. - .-. N — x. 

By taking an Enthymeme of the third order from the 
first Syllogism, and one of the first order from the second 
Syllogism of the first series, and putting them together 
in one expression, and, by taking an Enthymeme of 
the third order from the first Syllogism of the second 
series, out transposing the propositions so taken, and 
one of the second order from the second Syllogism of 
the same series, and putting them together in one ex- 
pression, we shall have the same Sorites, as before, in 
the two configurations, viz.: 



28 LOGIC AS A PURE SCIENCE. 

Regressive Sorites Progressive Sorites 

from the first series. from the second series. 

J - x; N — d; 

D - J 5 D - j ; 

JST — d; J — x; 

.-. N — x. .-. N — x. 

The conclusion of the first Syllogism in each series is 
held in the mind (otherwise there were no Enthymeme), 
but carried forward mentally, and employed as a pre- 
mise, still unexpressed, in connection with the Enthy- 
meme taken from the second. 

A Sorites considered as a complex expression as 
above shown is also called a Chain- Syllogism. 

§ 8. The middle premise (being the proposition D — j 
in which the minor-middle and major-middle terms are 
compared) will always be the second proposition in every 
Sorites, simple (as hitherto shown) or compound (as to 
which latter you will hereinafter be instructed) ; and by 
expressing it, in connection with the ultima, every 
Sorites may be still further abridged, thus : 

d- j; 

.-. N — x. 

All the four terms here appear, but each only once. 
Such an expression is in tJie form of an Entliymeme 
(but is not an Enthymeme, for that can have only three 
terms), and may properly be called ah Abridged Sorites. 

From the employment of the middle premise as the 
minor or major premise of the first Syllogism, I desig- 
nate Sorites (considered as complex expressions) minor 



SORITES. 



and major Sorites, respectively, for the purpose of classi- 
fication as hereinafter shown. Either may be regressive 
or progressive ; but we shall see that the proper division 
of Sorites is into regressives and progressives. 

Observe, that in all major Sorites, but in no minors, 
the premises constituting the Enthymeme of the third 
order taken from the first Syllogism, are transposed. 

§ 9. The Syllogisms of the two principal series (of 
Enthymemes of which the Sorites exhibited consist) 
are wholly in the first figure. But a little reflection 
will show that Sorites may also consist of Enthymemes 
taken from Syllogisms in any of the figures capable of 
combination in series, quantity and quality considered. 
And, as all Sorites may be abridged in the manner 
hereinbefore shown, it is also manifest that the range of 
possible abridged Sorites is limited to the number of 
possible combinations of two propositions composed of 
four terms, expressed in the same form as to the order 
of the terms throughout, but modified in respect to 
quantity and quality, as in the following scheme; and 
only such can be valid as are capable of being expanded 
into full Sorites, and from full Sorites into at least two 
series of Syllogisms. The propositions must be in one or 
another of the combinations shown by full lines in the 
scheme. 

D — i ; ^5= -z^r X — x. 



— J 




-_ J;^ 



X. 



n -^X. 



30 LOGIC AS A PURE SCIENCE. 

Considering the lines connecting the propositions, 
each as signifying "and therefore," there are sixteen dif- 
ferent combinations. But of these, only nine will be found 
to be valid, and they are symbolized by the same symbols 
as those of valid Enthymemes of the first order, as here- 
inbefore shown, and may be expanded into full Sorites 
(the supplied premises varying in the order of the terms 
as well as in quantity and quality), and from full Sorites 
into two, three, or four series of Syllogisms, with the 
middle premise as either the minor or the major premise 
of the first Syllogism of one or more series, except in two 
cases, which will be hereinafter noted. 

The number of valid full Sorites into which the nine 
abridged forms may be so expanded is one hundred and 
forty-four, of which one half are minors and one half 
majors, classified as such according to the combinations 
of the symbols of the abridged forms, as follows : 



Symbols. 


Minors. 


Majors 


A, A. 


1 


1 


A, E. 


4 


8 


A, I. 


16 


10 


A, 0. 


24 


24 


E, E. 


4 


4 


E, 0. 


10 


16 


I, I- 


4 


4 


I, 0. 


8 


4 


0, 0. 


1 


1 



72 ' 72 

The following synopsis exhibits all possible valid cate- 
gorical Sorites, in their abridged forms, as minors on the 



SOBITES. 31 

left-hand pages, and as majors on the right ; together with 
the premises by which they may be expanded into valid 
fnU Sorites, and the names of the moods in which they 
can be further and fully expanded into series of Syllo- 
gisms. They are arranged in the order A, I, E, O, of 
the symbols of the ultima. 

The abridged forms may be expanded into full Sorites 
by writing first, the first of the two supplied premises ; 
secondly, the middle premise; thirdly, the second of the 
two supplied premises ; and lastly, the ultima. 

Preceding the synopsis are given two series of schemes, 
by which the different ways in which abridged Sorites 
may be* expanded into full Sorites, and from full Sorites 
into series of Syllogisms, in all combinations of figures 
in which they are capable of being so expanded, may be 
clearly seen. The terms of the abridged Sorites are in 
capitals enclosed in circles connected by lines represent- 
ing the copulas of the propositions. The curved lines 
(considered as copnlas) above the propositions constitut- 
ing the abridged Sorites, in connection with those propo- 
sitions, indicate two expanded Sorites, and in connection 
also with the dotted straight line above, indicate two- 
series of Syllogisms ; and the lines below, two other 
expanded Sorites, and two other series of Syllogisms. 
The dotted straight lines show the unexpressed conclu- 
sions of the first Syllogisms, which in each case becomes 
one of the premises of the second. The modifications of 
the propositions of the abridged Sorites, in respect to 
quantity and quality, are indicated by the symbols above 
and below the lines representing their copulas respect- 
ively ; those above referring to the Sorites and Syllo- 



32 LOGIC AS A PURE SCIENCE. 

gisms indicated above, and those below, to those below. 
The modifications of the other indicated propositions 
are also in like manner signified. 

The symbols in connection with the lines are those 
only in which the Sorites and Syllogisms are valid in the 
figures shown. 

It is not meant that each symbol in connection with 
each other will yield a valid Sorites, but that each, in 
connection with some one or more of the others, will 
Ibe found valid. Thus, in the second scheme of minors, 
t,he maximus premise, A, will combine with the middle 
premise as E or 0, and E with A or I, but not other- 
wise. 

The designations of premises, written between paral- 
lel curved lines, refer to the propositions indicated by 
both lines ; the symbols and number of the figure being 
on the other side of each line, respectively. 

By marking all the lines with all the symbols, you 
will be able to make an exhaustive analysis of all possi- 
ble ways in which it may be attempted to frame simple 
Sorites. In view of the number given on the next page, 
you may think the attempt formidable, but you will find 
it not so much so as it will at first appear, if you but 
consider and apply to the symbols the rules of the syllo- 
gism before proceeding to test them. The lines above the 
propositions constituting the abridged Sorites are marked 
with all the symbols of the propositions respectively, as 
they may be employed in single simple syllogisms, as 
hereinbefore shown, but those below, not ; and if you 
first add to the latter the omitted symbols, making them 
to correspond with those above, you will find that such 



SOEITES. 33 

added symbols will, in all cases, yield no conclusion in 
the second of the Syllogisms, by reason of one or the 
other of the two faults, undistributed middle and illicit 
process of the major. If the remaining symbols be then 
added to each line, a violation of some one or more of 
the rules of the syllogism will be found in either the 
first or second Syllogism. 

The total number of the ways in which it may thus 
be attempted to combine the four symbols A, E, I, O, 
according to the schemes is eight thousand one hundred 
and ninety-two, that being the product of the number of 
ways (256) in which the four symbols may be combined 
(all the same, or partly the same, or all different), multi- 
plied by the number of combinations of propositions (4) 
indicated by each scheme, and again by the number of 
schemes (8)— (256 x 4 x 8 = 8192). 

The total number of valid Sorites without regard to 
their character as minor or major, or as regressive or pro- 
gressive, will be hereinafter found to be forty-four. 

By examining each . scheme, and comparing the 
Sorites and series of Syllogisms thereby indicated (those 
above with each other, and those below with each other), 
and by comparing each scheme with each of the others 
in all possible ways, the differences between, and corre- 
lations of, the several figures of the Syllogism and 
the two kinds of Sorites indicated by the schemes (that 
is, either minor or major), will also clearly appear, and 
the student cannot fail to be impressed with the har- 
mony and symmetry of pure reasoning, in all its varied 
possible forms of expression. 



34 



LOGIC AS A PURE SCIENCE. 



SCHEMES OF MINOR SORITES. 



FIRST SYLLOGISM IN FIRST FIGURE. 



Jstfig. Maj. 



£T e ^. 




(*}£&$(*) 






4.2. or E. 



A. I. or E. 



FIRST SYLLOGISM IN SECOND FIGURE. 
E.or O. 




©-" 




0-§^-0 




E. 



SORITES. 



35 



SCHEMES OF MAJOR SORITES. 

FIRST SYLLOGISM IN FIRST FIGURE. 

^ A. I;JE._or_q._ _ 

Min. p ^ A^or_E^__ lgt 




0xf^0 




A. or I. 



A. L or B. 



FIRST SYLLOGISM IN SECOND FIGURE. 

E. or O. 



_.. 1st fig. 




\ 



or J. 



A. or E. 



36 



LOGIC AS A PUEE SCIENCE. 



SCHEMES OF MINOR SORITES. 

FIRST SYLLOGISM IN THIRD FIGURE. 

I. or O. 



&***, 




FIRST SYLLOGISM IN FOURTH FIGURE. 

J._E._or_q._ _ 







(Vm 






). I ^] 



0l " T 



A- ° x 



I. or E. 



SORITES. 



37 



SCHEMES OF MAJOR SORITES. 

FIRST SYLLOGISM IN THIRD FIGURE. 

^ _J-_or_0._ 

Min. prem. 




i*^0 




r^^T^fx) 




A. or I. 



FIRST SYLLOGISM IN FOURTH FIGURE. 

_ I-_E._or O^ _ 
A.orJE. 







■ss^ 



Off 



A. or E. 



I. or E. 



38 



LOGIC AS A PUKE SCIENCE. 



Synopsis of all Possible Valid Forms of 

TOGETHER WITH ALL THEIR POSSIBLE MAGNUS AND MAXIMUS 
PREMISES, 









H H 




Sym- 
bols. 


Nos. 


Abridged Minor 
Sorites. 


Mood op 
First 


s^a gags 

« g 2 ° h o £ 

« 8 3 « « O 3 
Ph&i o oPk H O 


Mood of 
Second 






Middle Premise. Ultima. 


Syllogism. 


o fe r 1 




Syllogism. 


A, A. 


1 


D-j; ••• n-x. 


bArbAra 


J-x 


X— d 


barbArA 


A, I. 


2 


D— j ; •*• n— x. 


bArbAra 


J-x 


X-d 


a, A J. 1st fig. 




3 




a 


// 


n — d 


darll 




4 




( or, A, A, % \ 


// 


D— n 


j darAptl 
\ disAmls 




5 




bArbAra 


// 


d — n 


datlsl 




6 




brAmAntip 


X— d 


J-x 


dAtisI 




7 




dlmAris 


n— d 


u 


a 




8 




dArApti 


D-n 


a 


n 




9 




dlsAmis 


d — n 


a 


n 




10 




J bArbAra \ 
j or, A,A,i j 


J-n 


D-x 


j dArapfl 
1 dAtisI 




11 




bArbAra 


// 


d-x 


dlsam Is 




12 




a 


a 


X-d 


brAmantlp 




13 




a 


a 


x-d 


dlmarls 




14 




dArApli 


D-x 


J-n 


disAmls 




15 




dlsAmis 


d — x 


// 


a 




16 




brAmAntip 


X-d 


n 


n 




17 




dlmAris 


x-d 


n 


it 


1,1. 


18 


<*— j; ••• n— x. 


dArli 


J-x 


D— n 


disAmls 




19 


// 


dAtlsi 


D-n 


J-x 


dAtisI 




20 


„ 


dArli 


J-n 


D-x 


a 




21 


a a 


dAtlsi 


D-x 


J-n 


disAmls 



SORITES. 



39 



, Abridged Categorical Sorites. 



AND MOODS OF SIMPLE SYLLOGISMS IN WHICH THEY CAN BE FULLY 
EXPANDED. 



Stm- 

BOLS. 


Nos. 


Abridged Major 
Sorites. 


Mood of 
First 


a Premise 
First 

-LOGISM. 


"hi 

pj M O o 


Mood of 
Second 




1 


Middle Premise. Ultima. 


Syllogism. 
bArbAra 


d « f 
Sol" 




Stllogism. 


A, A. 


D-j; .-. N-x. 


N — d 


J-X 


bArbarA 


A, I. 


2 


D— j ; •■• n— x. 


j bArbAra j 
1 or, A, A,i f 


N— d 


J-x 


SA.a.I 
1 dAril 




, 3 


a n 


dArli 


n-d 


II 


" 




4 


a a 


dArApti 


D-n 


II 


a 




5 


a a 


dAtlsi 


d — h 


II 


ii 




6 


n n 


brAmAntip 


J-x 


D-n 


dimArls 




7 


a a 


dArApti 


D-x 


J-n 


a 




8 


n n 


dAtlsi 


d-x 


// 


ii 




9 


a n 


f bArbAra I 
j or, A, A, i f 


X-d 


// 


j bramAntlp 
j dimArls 




10 


ii n 


dArli 


x — d 


n 


n 




11 


it r 


brAmAntip 


J-n 


D-x 


dAril 


1,1. 


12 


d — j; .% n^x. 


dlsAmis 


D-n 


J-x 


dAril 




13 


a a 


dlmAris 


J-x 


D— n 


dimArls 




14 


a a 


dlsAmis 


D-x 


J-n 


it 




15 


a a 


dlmAris 


J-n 


D-x 


dAril 



40 



LOGIC AS A PTTEE SCIENCE. 



Synopsis of all Possible Valid Forms of 

TOGETHER WITH ALL THEIR POSSIBLE MAGNUS AND MAXIMU3 
PREMISES, 



Sym- 
bols. 


Nos. 


Abridged Minor 

Sorites. 

Middle Premise. Ultima. 


Mood of 

First 

Syllogism. 


H 

S H S 

_ < i 

«S 3 

0h£ o 

a .J 
t; o (h 


Minor or 

Major Premise 

of Second 

Syllogism. 


Mood of 

Second 

Syllogism. 


A, E. 


22 


D-j; .-. SF-X. 


cElArent 


J--X 


X-d 


celArEnt 




23 


// // 


cEsAre 


X— J 


// 


a 




24 


ti n 


cElArent 


dF-N 


X-d 


cAmenEs 




25 


n n 


cEsAre 


£-J 


a 


n 


E, E. 


26 


D-J; ... 2?_x. 


cAmEstres 


x-i 


X-d 


celArEnt 




27 


n ti 


cAmEnes 


X-d 


x-j 


cAmenEs 




28 


a a 


a 


X-d 


N-j 


celArEnt 




29 


a a 


cAmEstres 


* T -J 


X-d 


cAmenEs 


A, 0. 


30 


D-j; ••• n—X. 


cElArent 


J-X 


1ST— d 


j e, A, 0. 
"j 1st fig. 




31 


// /» 


n 


7/ 


n — d 


ferlO 




32 


// // 


\ or, E, A, o j 


I' 


D-n 


\f el Apt On 
| bokArdO 




33 


n n 


cElArent 


II 


d— n 


ferlsO 




34 


a a 


cEsAre 


X-J 


X-d 


j e, A, 0. 
( 1st fig. 




35 


a n 


a 


// 


n-d 


ferlO 




36 


a a 


\ or, E, A, o \ 


/; 


D-n 


Sf el Apt On 
1 bokArdO 




37 


a n 


cEsAre 


ii 


d — n 


ferls 




38 


a a 


cElArent 


J--N 


X-d 


j A, e, 0. 

i 4th fig. 




39 


a a 


cEsAre 


N-J 


a 


a 



SORITES. 41 

Abridged Categorical Sorites. (Continued.) 

AND MOODS OF SIMPLE SYLLOGISMS IN WHICH THEY CAN BE FULLY 

EXPANDED. 













B 


m 




Sym- 
bols. 


Nos. 


Abridged Major 

Sorites. 

Middle Premise. Ultima. 


Mood of 

First 
Syllogism. 


pq m xfi 

«53 

Oheq O 

a 02 
% 


Major or 

Minor Premii 

of Second 

Syllogism. 


Mood of 

Second 

Syllogism. 


A, E. 


16 


D-j; 


.: R—X. 


bArbAra 


X-d 


dF-X 


cElarEnt 




17 


// 


a 


" 


;/ 


5-J 


cEearE 




18 


// 


f 


a 


X-cl 


J-X 


camEnEs 




19 


n 


// 


u 


// 


X-J 


camEstrEs 




20 


n 


a 


cAmEnes 


J-X 


X— d 


cesArE 




21 


n 


a 


cAmEstres 


X-J 


a 


ii 




22 


n 


a 


cAmEnes 


J-X 


X-d 


cAmestrEs 




23 


it 


a 


cAmEstres 


X-J 


a 


ii 


E, E. 


24 


B--J 


; .-. X-X. 


cElArent 


X-d 


x-j 


cAmestrEs 




25 


/; 


// 


cEsAre 


x-j 


X-d 


cesArE 




26 


ii 


/' 


a 


N-j 


X-d 


cAmestrEs- 




27 


ii 


a 


cElArent 


X-d 


N-j 


cesArE 


A, 0. 


28 


D-j; 


.'. n-~X. 


J bArbAra \ 
\ or, A, A,i\ 


X— d 


J-X 


\fEriO 




29 


a 


// 


dArli 


n— d 


» 


ii 




30 


a 


a 


dArApti 


D-n 


// 


it 




31 


ii 


a 


dAtlsi 


d— n 


a 


ir 




32 


a 


a 


j bArbAra } 
\ or, A, A,i\ 


X — d 


X-J 


J .#, a, 
XfEstinO 




33 


a 


a 


dArli 


ii — d 


// 


ii 




34 


a 


n 


dArApti 


D-n 


II 


ii 




35 


a 


a 


dAtlsi 


d— n 


II 


ii 




36 


a 


a 


bArbAra 


X-d 


£— N 


j a, E, 0. 
\ 4th fig. 




37 


n 


a 


a 


7/ 


X-J 


1 2d fig. 




38 


a 


a 


a 


II 


n-^J 


bar Ok 



42 



LOGIC AS A PUKE SCIENCE. 



Synopsis of all Possible Valid Forms of 

TOGETHER WITH ALL THEIR POSSIBLE MAGNUS AND MAXIMUS 
PREMISES, 



Sym- 
bols. 


Nos. 


Abridged Minor 

Sorites. 

Middle Premise. Ultima. 


Mood of 

Fdsst 

Syllogism. 


Major Premise 

of First 

Syllogism. 


Minor or 

Major Premise 

of Second 

Syllogism. 


Mood op 

Second 

Syllogism. 


A. 0. 


40 


D-j ; .-. n 


-~X. 


fElAptm 


D-X 


J-ll 


bokArdO 




41 


" 


II 


b OkArdo 


cl-^X 


a 


n 




42 


n 


" 


fEsApo 


X-D 


a 


n 




43 


n 


II 


J bArbAra \ 
\ or, A, A, i S 


J -n 


D--X 


SfElapt On 
If ErisO 




44 


' ii 


II 


bArbA ra 


a 


d-wX 


bOJcardO 




45 


ii 


II 


j " ! 
| or, A, A,i f 


n 


X-D 


j/EsapO 

\ frEsis On 




46 


ii 


II 


brAmAntip 


N— d 


J-X 


/ErisO 




47 


ii 


II 


. dlmAris 


n— d 


a 


„ 




48 


ii 


" 


dArApti 


D-n 


» 


r 




49 


ii 


!' 


dlsAmis 


d — n 


a 


" 




50 


ii 


II 


brAmAntip 


X-d 


X-J 


frEsis On 




51 


ii 


n 


dlmAris 


ii— d" 


a 


a 




52 


ii 


II 


dArApti 


D-n 


>< 


ii 




53 


ii 




dlsAmis 


d-n 




n 


1, 0. 


54 


d— j; /. n 


-^X. 


fErlo 


J-X 


D-n 


bokArdO 




55 


'/ 


II 


fEstlno 


X-J 


a 




56 


7/ 


II 


fErlso 


B--X 


J-n 


n 




57 


" 


II 


frEsIson 


g-D 


a 


n 




58 


// 


II 


dAtlsi 


D— n 


J-X 


f ErisO 




59 


II 


n 


" 


a 


X-J 


frEsis On 




60 


II 


11 


dArli 


J_n D-X 


/ErisO 




61 


II 


ii 


ii 


" |X-D 
1 


frEsis On 



SORITES. 



43 



Abridged Categorical Sorites, (Continued.) 



AND 


MOODS OP SIMPLE SYLLOGISMS IN WHICH THEY CAN 


BE FULLY 




EXPANDED. 




Sym- 
bols. 


Nos. 


Abridged M&jor 

Sorites. 

Middle Premise. Ultima. 


Mood of 

First 
Syllogism. 


Minor Premise 

of First 

Syllogism. 

Major or 

Minor Premise 

of Second 

Syllogism. 


Mood of 

Second 

Syllogism. 


A, 0. 


39 


D-j; .-. n-^X. 


brAmAntip 


J-n 


&-X 


fEHO 




40 


// a 


a 


// 


X— D 


fEstin 




41 


a a 


cAmEnes 


J- X 


N-d 


j e, A, 0. 
1 2d fig. 




42 


n a 


» 


a 


n — d 


festlnO 




43 


a a 


" 


a 


D-n 


fesAp 




44 


a a 


ii 


a 


d — n 


fresls On 




45 


a a 


cAmEstres 


3E-J 


N— d 


S e, A, 0. 
1 2d fig. 




46 


II If 


. a 


a 


n — d 


festln 




47 


it ii 


" 


a 


D-n 


fesApO 




48 


II II 


ii 


a 


d — n 


fresls On 




49 


II ti 


j cAmEnes { 
( or, A, E, o f 


J-N 


X-d 


}A,e,0 
1 bArokO 




50 


ii n 


j cAmEstres ) 
| or, A, E, o f 


BF— J 


n 


SA,e,0 
j bArokO 




51 


n it 


bArOko 


n-^J 


n 


ii 


I, 0. 


52 


d-j; ••• n-~X. 


dlsAmis 


D— n 


J-X 


fEHO 




53 


It II 


it 


// 


X--J 


fEstin 




54 


It II 


dlmAris 


J-n 


D-X 


fEHO 




55 


II II )- 


n 


t> 


X-D 


fEstin 



44 



LOGIC AS A PURE SCIENCE. 



Synopsis of all Possible Valid Forms of 

TOGETHER WITH ALL THEIR POSSIBLE MAGNUS AND MAXIMUS 
PREMISES, 











H 


H 




Sym- 
bols. 


Nos. 


Abridged Minor 

Sorites. 
Middle Premise. Ultima. 


Mood of 

First 
Syllogism. 


w JS s 
n «S3 

K N 
o & H 
fc, o h 

<q CO 
8 


Minor or 

Major Premis 

of Second 

Syllogism. 


Mood of 

Second 

Syllogism. 


E, 0. 


62 
63 


B--J ; .-. n-wX. 


cAmEstres 
n 


x-j 

II 


N— - d 
n — d 


e,J, 0. 1st fig. 
ferlO 




64 




\ or, A, E, o \ 


II 


D-n 


\f el Apt On 
| bokArdO 




65 




cAmEstres 


II 


d — n 


ferlsO 




66 




cAmEnes 


X-cl 


N-j 


e,4,0. lstfig. 




67 




ii 


a 


n — i 


ferlO 




68 




\ or, A, E,o ( 


n 


J-n 


\f el Apt On 
\ bokArdO 




69 




cAmEnes 


n 


j- n 


ferlsO 




70 




a 


N— d 


x-j 


"J. a 0.4th fig. 




71 


/; /; 


cAmEstres 


N-j 


X-d 


^l,e, 0.4th fig. 


0, 0. 


72 


d-~J;.\ n-vX. 


bArOTco 


X-3 


D-n 


bokArdO 



SORITES. 



45 



Abridged Categorical Sorites. (Concluded.) 

AND MOODS OF SIMPLE SYLLOGISMS IN WHICH THEY CAN BE FULLY 
EXPANDED. 











H 


a 




Sym- 
bols. 


Nos. 


Abridged Major 
Sorites. 


Mood of 
First 


« r J 


.JOR OR 
R PREMI! 

Second 

jLOGISM. 


Mood op 
Second 




56 


Middle Premise. Ultima. 


Syllogism. 


® pi £ 


'=' g OcO 


Syllogism. 


E, 0. 


B-- J ; .-. n-^X. 


J cElArent 1 
j or, E, A, o ( 


X-d 


x-j 


SA,e.O 
| bArokO 




57 




fErlo 


n — d 


/; 


ii 




58 




fElAyton 


D-n 


// 


it 




59' 




fErlso 


d — n 


/r 


ii 




60 




j cEsAre \ 
( or, j?, .4, o f 


s-j 


X-d 


$A,e,0 
l bArokO 




61 




fEstlno 


n — j 


// 


ii 




62 




fEsApo 


J-n 


;/ 


n 




63 




frEsIson 


J- n 


• 7/ 


ii 




64 




cEsAre 


x-j 


X— d 


M,6>.2dfig. 




65 




a 


:/ 


n — d 


/erfjfl 




66 




1! 


it 


D-n 


fesAp 




67 




II 


n 


d — n 


f reals On 




68 




cElArent 


X-d 


N-j 


e,A,0 2d fig. 




69 




a 


n 


n-j 


/es*/« 




70 




a 


a 


J-n 


fesApO 




71 




a 


a 


j- n 


fresIsOn 


0, 0. 


72 


d-wJ ; .-. n-wX. 


bOkArdo 


D-n 


x-j 


bArokO 



46 



LOGIC AS A PURE SCIENCE. 



§ 10. By examining the foregoing synopsis and testing 
the same, it will be found that 

If the major-middle term (predicate of the middle premise) he 
the middle term of the first Syllogism, then if the Sorites be 







Minor ; but if it be 




Major ; 






and the configurations of 
the Sorites, whether re- 
gressive or progressive, or 
both, and the number of 
each, will be as follows ; 




« ■^•» 

s § - 

B 


and the configurations of 
the Sorites, whether re- 
gressive or progressive, or 
both, and the number of 
each, will be as follows : 


1 


1 


Beg., 


6 














1 


3 


Reg., 


6 


Prog. , 


6 


2 


2 


Peg.. 


6 


Prog., 6 


1 


4 






Prog., 


6 


2 


4 


Reg.. 


4 






2 


1 


Reg., 


6 






4 


1 






Prog, 


4 


2 


3 


Reg., 


6 






4 


2 


Reg.. 


3 Prog, 


6 


2 


4 






Prog. , 


4 


4 


4 


Reg, 


4. 







hit if the 'minor-middle term (subject of the middle premise) 
be the middle term of the first Syllogism, then, if, as sec- 
ondly above, the figures of the Syllogisms may be, and the 
configurations of the Sorites and the member of each will be 
as follows : 



Minor : 



Major 















1 


1 






Prog. , 


6 


3 





Reg., 


6 


Prog., 


6 


1 


2 


Reg., 


6 


Prog., 


6 


3 


4 






Prog., 






1 


4 


Reg, 


6 






4 


1 


Reg, 


3 






3 


1 






Prog., 


6 


4 






Reg., 


6 


Prog., 


4 


3 


2 






Prog., 


6 


4 


4 




39 


Prog., 


4 
33 


3 


4 1 


Reg, 


3 
32 




40 



Total minors, 72 ; Total majors, 72 ; 

Grand total, L44. 



SORITES. 47 

But, by a careful examination of the synopsis, it will 
be found that fifty-six of the Sorites are both minors 
and majors. That number must therefore be deducted 
from the grand total, leaving eighty-eight different 
forms. 

Each of the four figures occurs as the figure of the 
first Syllogism in both minor and major Sorites ; but the 
second does not occur as the figure of the second Syllo- 
gism in minors, nor the third in majors. With these 
exceptions, all the figures occur also as figures of the 
second Syllogism. 

The following is a synopsis of all the eighty-eight 
possible "forms of valid simple Sorites arranged according 
to their configurations, regressives on the left-hand pages, 
and progressives on the right, and without regard to their 
being either minor or major, but showing in the columns 
on the left-hand side of each page, the moods of the 
Syllogisms in respect to which they are minors, and on 
the right, those in respect to which they are majors. 

There will be found on the pages of regressives, seven- 
teen, and on the pages of progressives, fifteen, in which 
the moods are only on one side, leaving twenty-seven 
regressives and twenty -nine progressives in which the 
moods are on both sides, and which together make the 
fifty-six alike on both sides of the preceding synopsis, 
as above stated. Two, namely, lS"os. 25 and 38, are the 
exceptions hereinbefore referred to. ISTo. 25 is a minor 
Sorites only, and ]STo v 38 a major Sorites only, in both 
configurations. 

As before, they are arranged in the order A, I, E, O 
of the symbols of the ultima. 



48 



LOGIC AS A PURE SCIENCE. 



Synopsis of all Possible Valid Forms 



Series of Syllogisms 
in which the Middle 


GO 

n 

o 


Regressive Configuration. 


Series of Syllogisms 
in which the Middle 


Premise 


is Minor, 

Maximtts 

Major of 


1 

o 

o 

1 


Maximus 
Premise. 








Premise is Major, 


and the 
Premise 
the first 


Middle 
Premise. 


Magnus 
Premise. 


Ultima. 


and the Maximus 
Premise Minor of 
the first. 


bArbAra 


barbArA 


J-x D-j 


X-d 


/.X-x 






bArbAra 


la, A, I. I 
1 1st fig. f 


2 


J-x D-j 


X-d 


.*. 11 — X 






bArbAra 


darll 


3 


J-x D-j 


n — d 


.*. 11 — X 






bArbAra 
or, A, A, i 


darAptl | 
disAmls j 


4 


J-x D-j 


D-n 


.•. n — x 


(2) (1) 
brAmAntip 


(3) (4) 
dimArls 


bArbAra 


datlsl 


5 


J-x 


D-j 


d— n 


.*. n — X 






dArli 


disAmls 


6 


J-x 


d-j 


D— n 


.-. 11 — X 


dlmAris 


dimArls 


dArApti 


disAmls 


7 


D-x 


D-j 


J— n 


.". 11 — X 


dArApti 


dimArls 


dAtlsi 


disAmls 


8 


D-x 


d-j 


J -ii 


.•. 11 — X 


dlsAmis 


dimArls 


d Is Amis 


disAmls 


9 


d— x 


D-j 


J-n 


.". 11 — X 


dAtlsi 


dimArls 


brAniAntip 


disAmls 


10 


X-d 


D-j 


J-n 


.'. 11 — X 


j bArbAra 
\ or, A,A,i 


bramAntlp 
dimArls 


dlmAris 


disAmls 


11 


X — (1 


D-j 


J-n 


.*. n — x 


dArB 


dim A rls 


cElArent 


celArEnt 


12 


J-X 


D-j 


X — d 


.-.X— X 


cAmEnes 


cesArE 


cEsAre 


celArEnt 


i:; 


X-J 


D-j 


X— d 


.-.X-X 


cAmEstres 


cesArE 






14 


X-d 


D-j 


X-J 


.-.X-X 


bArbAra 


camEstrEs 






15 


X-d 


D-j 


J-X 


.-.X-X 


bArbAra 


camEnEs 


cAmEnes 


celArEnt 


16 


X-d 


D--J 


N-j 


.-.X-X 


cElArent 


cesArE 


cAmEstres 


celArEnt 


17 


x-j 


D-J 


X-d 


.-.X-X 


cEsAre 


cesArE 



SORITES. 



49 



of Simple Categorical Sorites. 



Series of Syllogisms 




IN "WHICH THE MIDDLE 




Premise is Minor, 




AND THE 


Magnus 




Premise 


Major of 




THE FIRST. 






brAmAntip 


dAtisI 




dlmAris 


dAtisI 




dArApti 


dAtisI 




dlsAmis 


dAtisI 




dAtM 


dAtisI 




bArbAra 
or, A, A., i 


dAraptl ) 
dAtisI ) 




dArli 


dAtisI 




bArbAra 


dlsamls 




bArbAra 


brAmantlp 




bArbAra 


dlmarls 




cEsAre 


cAmenEs 




cElArent 


cAmenEs 




cAmEstres 


cAmenEs 




cAmEnes 


IcAmenEs 





Progressive Configuration. 



Magnus Middle Masimus 
Premise. Premise. Premise. 



Ultima. 



]S[_d D— i 

! 

X— d D— j 

n _d ; D— j 

D— n D— j 

d-n D-j 
D— 11. | d— j 

J -ii 
J -ii 
J— n 

J -n 



J- ii 

X — d 
X— d 
X-J 
J— S 



j_ x .-.X-x 

J — X .*. 11 — X 
J — X /. 11 — X 

J_ x .-. n _x 

J — X .". 11 

11 — X 



Series of Syllogisms 
in which the Middle 
Premise is M a job, 
and the Magnus 
Premise Minor of 
the first. 



11 X 



J-x 
D- D— x 

d_j D — x 

B_j j d— x I/, n — x 

i 

D— j |x— d U ii— x 



D-j 

D-j 
D-j 

D-j 

D-j 



X-j B--J 



X-d 



B--J 



x-d 

J-X 

X-J 
X-d 
X-d 
X — d 
X-j 



(2) (1) 
bArbAra 



bArbAra 
or, A, A, i 



dArli 

dArApti 

dAtlsi 
dlsAmis 

brAmAntip 

dlmAris 



.-.X-X 
.-.X-X 
X-X 
X-X 
X— X 
X-X 



(3) (4) 
bAfbarA 



A, a. I 
dAril 



dAril 

dAril 

dAril 
dAril 

dAril 

dAril 



bArbAra 

bArbAra 

c Am Est res 

cAmEnes 

cEsAre 

cElArent 



cElarEnt 

cEsarE 

cAmestrEs 

cAmestrEs 

cAmestrEs 

cAmestrEs 



50 



LOGIC AS A PUKE SCIENCE. 



Synopsis of all Possible Valid Forms 



Series of Syllogisms 
in which the Middle 


a 
o 

o 


Regressive Configuration. 


Series of Syllogisms 
in which the middle 


Premise 


is Minor, 
Maximus 


ft 
o 






Premise is Major, 


AND THE 










and the Maximus 


Premise 


Major of 


m 


Maximus 


Middle 


Magnus 


Ultima. 


Premise Minor of 


Tun FIRST 




o 

18 


Premise. 


Premise. 


Premise. 


THE FIRST. 


cElArent 


j e, A, 0. | 
1 1st fig. f 


J-X 


D-i 


X-d 


.-. n-^X 


(2) (1) 
cAmEnes 


(3) (4) 
\\e,A,0. 
j 1 2d fig. 


cElArent 


ferlO 


19 


J-X 


D-j 


n— d 


.-. n-^X 


xAmEnes 


festlnO 


cElArent 
or, E, A, o 


f el Apt On 1 
bokArdO | 


20 


J-X 


D-J 


D— n 


.-. n-^X 


cAmEnes 


fesAp 


cElArent 


ferlsO 


21 


J— X 


D-J 


d-n 


.-. n—X 


cAmEnes 


'fresIsOn 


fErlo 


bokArdO 


22 


J-X 


d-j 


D— n 


.-. n-^X 






fElApton 


bokArdO 


23 


B-— X 


D-J 


J-n 


.-. ri-_X 






fErlso 


bokArdO 


24 


B--X 


d-j 


J -n 


.-. n-^X 






bOkArdo 


bokArdO 


25 


d-^X 


D-j 


J-n 


A n-_X 






fEsApo 


bokArd 


26 


X-D 


D-j 


J-n 


A n-^X 






frEsIson 


bokArdO 


27 


3E— D 


d-j 


J— n 


A n-^X 






cEsAre 


j e, A, 0. \ 
1 1st fig. f 


28 


X— J 


D-j 


X-d 


•. n-^X 


cAmEstres 


j e, A, 0. 
I 2d fig. 


cEsAre 


ferlO 


29 


X-J 


D-j 


n — d 


\ n-~X 


cAmEstres 


jest In 


cEsAre 
or, E, A,o 


f el Apt On) 
bokArdO j 


30 


X-J 


D-j 


D-n 


\ n-^X 


cAmEstres 


fesAp 


cEsAre 


ferlsO 


31 


X-J 


D-j 


d — n 


•. n-^X 


cAmEstres • 


fresIsOn 


fEstlno 


bokArdO 


32 


X-J 


d-j 


D-n 


\ 11-wX 







SORITES. 



51 



of Simple Categorical Sorites. (Continued,) 



Series of Syllogisms 
in which the meddle 
Premise is Minob, 
and the Magnus 
Premise Major of 
the first. 



brAmAntip fErisO 

dlmAris fErisO 

dA rApti fEris 

dlsA mis : fEris 

dAtlsi fErisO 

bArbAra fElaptOn I 

or, A, A, i fErisO f 

dArli \ fErisO 

bArbAra bOkardO 



bA rbA ra fEsap 
or, A, A, i frEsisOn 



dArli "frEsis On 

brAmAntip frEsis On 

dlmAris frEsis On 

dArApti ■ frEsis On 

dlsAmis frEsis On 
dAtlsi \ frEsis On 



18 
19 

20 

21 
2 2 

23 

24 
25 

26 

27 

28 

29 

30 

31 
32 



Progressive Configuration. 



Magnus 
Premise. 



X — d 

n— d 
D— n 

a- n 

D-n 

J -n 

J- n 
J-n 

J — n 

J-n 

N— d 

n- d 

D-n 

d— n 
D — ii 



Middle Maximus 
Premise. ' Premise 



D 

D 

D 

D- 

(1- 

D 

d- 
D 

D 

d- 

D- 

D- 
D- 

D- 

d— 



J — X 



, Ultima. 



11-^X 



-j|j-X 
-j ,dF-X 



-J 



dF-X 



-3 i d -* 
-.] ,'s-D 

-i |x-d 
-j 3E-J 

-j jX-J 

-i x-j 

-j X-J 



.•. n- 

.•. n- 

.'. ii- 
.*. n- 

.\ n- 

.'. n- 
.•. ii- 

,\ n- 

;". 11- 



X-J 



'. 11- 

•. 11- 

•. 11- 

*. n 



_X 

_x 

_x 

.X 

.X 
.X 

.x 

.X 

.X 

.X 
.X 

-^x 



Series of Syllogisms 
in tvhich the Middle 
Premise is Major, 
and the Magnus 
Premise Minor of 
the first. 



(2) CD (3) (4) 
f bArbAra E, a, 
1 or, A,A,i fEriO 



dArli fEriO 

dArApti fEriO 

dAtlsi \fEriO 

dlsAmis fEriO 

brAmAntip fEriO 

dim A) is fEriO 

brAmAntip \fEstinO 

dlmAris fEstinO 

I bArbAra 
I or. A, A,i 

dArli fEstinO 

dArApti f Estiii O 



E,a, O 
fEstin O 



dAtlsi 
dlsAmis 



fEstinO 
fEstin 



52 



LOGIC AS A PUKE SCIENCE. 



Synopsis of all Possible Valid Forms 



Series of Syllogisms 
in which the middle 


R 
o 
o 


Regressive Configuration. 


Series of Syllogisms 
in which the middle 


Premise is Minor, 


F, 

ft 
o 








Premise 

AND TH] 


is Major, 


AND THE MAXIMUS 










] Maximds 


Premise Major op 




Maximns 


Middle 


Magnus 


Ultima. 


Premise 


Minor of 


THE FIRST. 


c 

33 


Premise. 


Premise. 


Premise. 


THE FIRST. 


cAmEnes 


j e, A, 0. I 
I 1st fig. f 


X-d 


fr-J 


*-J 


.-. ii-^X 


! 

(2)(1) 

cMArent 


(3) (4) 
j e, A, 0, 

{ 2d fig 


cAmEnes 


ferlO 


34 


X-cl 


B~-J 


n ~ D 


.-. n-^X 


cElArent 


festlnO 


cAmEnes 
or, A, E, o 


felAptOn ) 
bokArdO j 


35 


X-d 


B"-J 


J-n 


.-. n-^X 


cElArent 


fesAp 


cAmEnes 


ferlsO 


36 


X-d 


D--J 


D- n 


.-. n-^X 


cElArent 


fresls On 






37 


X-d 


D-D- 


ff— J 


.-. n-^X 


bArbAra 


\ a, E, 0. 
"1 2d fig. 






38 


X-d 


D-D 


n-^J 


.-. n-^X 


bArbAra 


bar Ok 






39 


X-d 


D-j 


J-X 


.-. n-^X 


bArbAra 


j a, E, 0. 
1 4th fig. 


cAmEstres 


}e,A,0. \ 
\ 1st fig. f 


40 


x-j 


D-J 


X T -d 


/. n-_X 


cEsAre 


j e, A, 0. 
1 2d fig. 


cAmEstres 


ferlO 


41 


x-j 


D--J 


n — d 


.-. n-^X 


cEsAre 


festlnO 


cAmEstres 
or, A,E,o 


felAptOn I 
bokArdO \ 


42 


x-j 


B-J 


D-n 


.-. n-^X 


cEsAre 


fesAp 


cAmEstres 


ferlsO 


43 


x-j 


B--J 


d — n 


.-. n-^X 


cEsAre 


f?-esls On 


bArOko 


bokAriO 


44 


X-j 


cl-_J D-n 


.-. n-^X 







SOEITES. 



53 



of Simple Categorical Sorites. (Concluded.) 



Series of Syllogisms 
in which the Middle 


R 

o 
o 


Progressive Configuration. 


Series of Syllogisms 
in which the Middle 


Premise 


is Minor, 
: Magnus 


o 






Premise 

AND TH 


is Major, 


AXD TH] 










e Magnus 


Premise 


Major of 


03 


Magnus 


Middle 


Masimus 


Ultima. 


Premise 


Minor of 


the first 




o 

33 


Premise. 


Premise. 


Premise.* 


THE FIRST. 


cArnEstres 


j A, e, 0. t 
1 4th fig. f 


N-j 


B--J 


X-d 


.-. n-^X 


(2)00 

j cEsAre 
I or,E,A.o 


(3) (4) 
A, e, 
bArokO 






34 


11 — j 


B--J 


X— <1 


.-. n-^X 


fEstlno 


bArokO 




■ 


35 


J-n 


B--J 


X — d 


.-. n-^X 


fEsApo 


bArokO 






36 


j- n 


B--J 


X— d 


.-. n-^X 


frEsTson 


bArokO 


cEsAre 


< A, e, 0. 

1 4th fig. 


37 


SJ.-J 


D-J 


X-d 


.-. n-^X 


j cAmEstre- 
1 or, A,E,o 


A. e. 
bArokO 






38 


n-^J 


D-J 


X-d 


.-. n-~X 


bArOko 


bArokO 


cElArent 


1 A, e, 0. 
\ 4th fig. 


39 


J-N 


D-J 


X-d 


.-. n-^X 


j c Am E lies 
\ or, A,E,o 


A, e, 
bArokO 


cAmEnes 


1 A, e, 0. 

1 4th fig. 


40 


N — d 


B--J 


x-j 


.-. 11-.X 


j cElArent 
1 or, E,A.o 


Li,e, 
bArokO 






41 


n — d 


B--J 


x-j 


.-. n-wX 


fErlo 


bArokO 






[-> 


D— n 


D--J 


x-j 


.-. n—X 


fElApton 


bArokO 






43 


cl — 11 


B--J 


x-j 


.-. n-^X 


fErlso 


bArokO 






44 


D-n 


d-^J 


x-i 


.-. n-^X 


bOkArdo 


bArokO 



S4 LOGIC AS A PUEE SCIENCE. 

§ 11. The number of forms of valid Sorites, shown in 
the foregoing synopsis, is eighty-eight, forty-four on each 
side ; but a comparison of them, line by line, read across 
both pages of the synopsis, will show that, considered 
with respect to the propositions of which they are com- 
posed, without regard to the order of their statement, 
there are but forty-four ; the first and third propositions 
in the regressive configuration changing places in each 
case, and becoming respectively third and first in the 
progressive throughout the whole series, the middle 
premise and ultima being the same in each case on both 
sides throughout. They are numbered from one to forty- 
four, on each side, to correspond. 

To one or another of these forms, EVERY valid argu- 
ment (expressed categorically) involving four terms, or, 
as will be hereinafter shown, involving any greater num- 
ber of terms, MUST BE conformed. 

§ 12. The moods, as determined by the quantity and 
quality of the propositions (indicated by their symbols), 
are twenty in number, of which fourteen occur in both 
configurations, three in the regressive only, and three in 
the progressive only. 

The following table shows them, arranged in the order 
A, I, E, O of the symbols of the ultima, with their num- 
bers in each configuration, as in the synopsis, repeated 
where they are both minor and major. The symbols are 
in capitals in the synopsis, the first two in the columns 
of Syllogisms, on the right-hand side of each page 
{majors) being transposed, as previously stated, and as 
shown by the figures over those columns. 



SORITES 



55 



Moods of Sorites. 





Nos. in Regressive Configuration. 


Nos. in Progressive Configuration. 


Symbols. 










Minors. 


Majors. 


Minors. 


Majors. 


A, A, A, A. 


1. 






1. 


A, A, A, I. 


2, 4, 7, 10. 


4, 7, 10. 


2, 4, 7, 10. 


2 4 7 


A, A, I, I. 


3, 5. 




9, 11. 




I, A, A, I. 


9, 11. 


9, 11. 


3, 5. 


3, 5. 


A, I, A, I. 


6, 8. 


6, 8. 


6, 8. 


6, 8. 


E, A, A, E. 


12, 13. 


12, 13. 


14, 15. 


14, 15. 


A, A, E, E. 




14, 15. 




12, 13. 


A, E, A, E. 


16, 17. 


16, 17. 


16, 17. 


16, 17. 


E, A, A, 0. 


1 18, 20, 23, ) 
1 26, 28, 30. f 


18, 20, 28, 30. 


37, 39. 


37, 39. 


A, A, E, 0. 




37, 39. 


( 18, 20, 23, 1 
{26, 28, 30. j 


j 18, 20, 23, ) 
1 26, 28, 30. (" 


E, A, I, 0. 


19, 21, 29, 31. 


19, 21, 29, 31. 






I, A, E, 0. 






19, 21, 29, 31. 


19, 21, 29, 31. 


E, I, A, 0. 


22, 24, 27, 32. 








A, I, E, 0. 






22, 24, 27, 32. 


22, 24, 27, 32. 


0, A, A, 0. 


25. 






38. 


A, A, 0, 0. 




38. 


25. 




A, E, A, 0. 


33, 35, 40, 42. 


33, 35, 40, 42. 


33, 40. 


33, 35, 40, 42. 


A, E, I, 0. 


34, 36, 41, 43. 


~34, 36, 41, 43. 






I, E, A, 0. 








34, 36, 41, 43. 


A, 0,A,0. 


44. 






44. 



56 LOGIC AS A PUKE SCIENCE. 

It is manifest that it would be a very difficult thing 
to classify Sorites in figures, according to the positions 
of the terms, and to devise names for the moods, anal- 
ogous to those of simple Syllogisms ; and, if it should 
be accomplished, the figures and names of the moods 
would be extremely burdensome to the memory. The 
different forms can Ibe much more readily referred to 
by their numbers and the names of the configurations, 
as adopted, than by their symbols, or any names that 
could be devised for them. They will be hereinafter so 
referred to. 

By counting the series of Syllogisms on the left 
(minors) and right (majors) of the synopsis in each con- 
figuration, there will be found to be : 



ressives. Progressives. 

Minors. Majors. Minors. Majors. 

39. 82. 33. 40. 

corresponding to the numbers shown in the table on 
page 46. 

§ 13. Sorites, in the regressive configuration, may be 
expanded into series of Syllogisms in all combinations of 
figures, except those of the third and first, and third and 
second ; and those in the progressive configuration, in all 
combinations, except those of the second and first, and 
second and third. 

Such of them as can be expanded wholly in the first 
figure, are the only perfect forms. The series of Syllogisms, 
in which they can be so expanded, occur in the synopsis 
only on the left side of the regressives (minors), and on 
the right side of the progressives (majors) ; and the first 



SOEITESo 57 

figure occurs as the figure of the second Syllogism only 
on the same sides. Moods Nos. 10, 11, 15, 26, 27, 35,, 
36, " and 39 cannot be expanded directly (that is, without 
conversion) except by the aid of the fourth figure ; a 
fact which may tend in some measure to relieve that 
figure from the odium which has been cast upon it. 

§ 14. There is a very remarkable and wonderful anal- 
ogy between the forms of reasoning and the two simplest 
forms of geometrical figures, plane and solid (with plane 
surfaces) ; an analogy which is evidently something more 
than merely fanciful. 

The Syllogism of logic and the triangle of geometry, 
and the Sorites and tetrahedron are, respectively, similar. 

The triangle consists of three points, equivalent to 
the three terms of the Syllogism, connected by three 
lines, which answer to the copulas of the propositions. 
No plane surface can be represented by less points and 
lines, no argument by less terms and propositions. By 
means of the former, with the aid of the latter, all phys- 
ical relations in space are determined, not only on the sur- 
face of the earth and within it, from those of the smallest 
subdivision to those of continents and oceans, but also in 
the heavens to the remotest star-depths, so far as the 
stars can be brought under observation ; by the latter 
all relations are determined, not only of physical things, 
but also of the metaphysical and immaterial. But the 
analogy does not end here. In its very practical con- 
struction the triangle produces the equivalent of a per- 
fect Syllogism in Barbara. If we are at any point, JST, 
on the surface of the earth, from which we can see 



58 LOGIC AS A PUKE SCIENCE. 

another point, J (either on the earth or in the heavens), 
which is inaccessible, and the distance to which we can- 
not therefore directly measure, we may select another 
point, D (either on the earth or its orbit), which is acces- 
sible, and from which the point, J, may also be seen ; 
and first, carefully observing the directions from IS" to J, 
and from ~N to D, and thus determining the angle, we may 
then proceed to measure the distance between N and D 
in a straight line. The line thus laid down is equivalent 
to the first proposition, !N" — d, with which we set out in 
§ 2 of this chapter. Arrived at D, we may then observe 
the direction from D to J, and determine the angle, and 
then, by means of the elements thus obtained, we may 
determine the distance in a straight line from D to J, and 
from ~N to J. The lines thus drawn, or supposed to be 
drawn, are the equivalents of the second proposition, 
D — j, with which we set out, and of the conclusion to be 
deduced from it and the first proposition, N — d, when 
put forth as premises of a Syllogism, namely, N — j. 

The tetrahedron is the simplest form in which any 
solid with plane surfaces can be included, and is the 
analogue of the Sorites. Its four points answer to the 
four terms, its four planes (each in the form of a triangle) 
bounded by six lines (each being a boundary of two 
planes) to the four Syllogisms of the two principal 
series ; each series with its six propositions. Each plane 
connects three points, each Syllogism three terms. Each 
of the four points is excluded from one of the planes, 
each of the four terms from one of the Syllogisms. 
To illustrate by means of geometrical figures : 
If we take a piece of card-board and, having cut it in 



SOEITES. 59 

the form of an equilateral triangle, inscribe therein 
another equilateral triangle, the lines of which terminate 
in the middle of the lines of the exterior one, and mark 
all the angles with letters, as follows : 




we may then fold the card-board backward on the lines 
of the inscribed triangle so as to bring together the three 
points, X, X, X, and then fastening together the edges 
of the card -board so brought together, we shall have a 
regular tetrahedron, the very embodiment of a simple 
Sorites. Looked at from our present stand-point, we 
shall see only the inscribed triangle No. 3, and having its 
angles marked with the letters N, D, and J. The other 
triangles and the point X will not be seen. Turning the 
figure about, so as to bring its planes before us in the 
order in which they arfe numbered, and considering them 
in two series of two each, we shall rind them as follows, 
beginning at the right hand with the first series, and 
reading backward, but from left to right, in the second. 



60 



LOGIC AS A PURE SCIENCE 
First series. 
D\ 





Second series. 





Observing that the letters at the apices of the tri- 
angles are the middle terms of the Syllogisms of the two* 
principal series hereinbefore shown, and considering the 
lines of the triangles as copulas connecting the terms of 
propositions, and the lines at the bases as indicating con- 
clusions, and beginning with the first series of triangles 
at the right hand and regressing, we can read as follows : 

Because D is J and J is X, therefore D is X ; and because N is D and 
D is X, therefore N is X, 

and then going to the second series, and beginning at the 
left hand and progressing, we can further read : 

Because N is D and D is J, therefore N is 'J ; and because N is J and 
J is X, therefore N is X. 

The correspondence between the triangles and the 
Syllogisms is exact throughout, except that the premises 



SORITES. 



61 



in the latter are transposed, bnt the order of statement 
of the premises is a matter of no consequence, the terms 
determining their character. 

The middle terms D and J may, of course, be trans- 
posed in our original illustration, and in such case the 
numbers 2 and 4 would also have to be transposed, and 
the positions of all the letters and the numbers in tri- 
angles 1 and 3, relatively to the whole figure, would 
also require to be changed. The first series of triangles 
would then read forward and the second backward, but 
the series of Syllogisms would remain the same, the 
first regressive, and the second progressive. 

The four triangles may also be exhibited in the 
following form : 




and may be folded on the interior lines with like result 
as before. 

But the Sorites is superior to its analogue, the tetra- 
hedron, in this, that its ultimate conclusion is reached by 
either process, regressive or progressive, but both are 
required to complete^ the tetrahedron. This will be 
apparent by the consideration of the two following 
forms. 



62 



LOGIC AS A PURE SCIENCE. 





If, in the first, beginning with JS", we successively 
reach by investigation the points D, J, and X, and then 
commence to reason with the propositions which we 
enounce as the results of our investigation, we may by 
two Syllogisms, of which the two Completed triangles 
1 and 2 are analogues, arrive at the ultimate conclusion. 
But if, in the second, by the same process of investiga- 
tion we reach only to the point J, and then commence 



SORITES. 63 

to reason, we frame our first Syllogism, of which the 
triangle 3 is the analogue, resulting in the conclusion 
that 1ST — J. We are thereupon, if we would advance 
further, obliged to resume investigation, and through it 
reach out to X, and are thence enabled to frame the 
second Syllogism, of which the triangle 4 is the analogue, 1 
arriving at the same ultimate conclusion. But in either 
case the tetrahedron is incomplete, and can only be com- 
pleted by the union of the two. Each figure is the com- 
plement of the other, required to make the perfect figure, 
shown in our first illustration. 

But again, the two different processes, regressive and 
progressive, in respect to argumentation by Syllogisms, 
are analogous to the two possible combinations of the 
two processes by which we may determine the length of 
the concluding line with which we enclose a triangle. 

Leaving X, and going to D, we observe the direction 
in which we are traveling, and measure the distance 
traveled. Then observing the direction from D to J, 
and thus determining the angle, we go on from D to J, 
measuring the distance. If we then stop, we may, by 
the three elements thus obtained, viz., the two lines and 
the included angle, determine the distance and direction 
from X to J ; then, having this distance and direction, 
and observing the direction of X from J, we go back to 
N, and observe its direction from X, and determine the 
angles, and then with the three elements thus secondly 
obtained, viz., the two angles and the included line from 
X to J, we may determine the distance from X to X. 
This is analogous to the progressive process. 

But if, after reaching J, without stopping to de- 



64 LOGIC AS A PURE SCIENCE. 

termine its distance from X, we observe the direction 
therefrom to X, as in triangle 1, and going back to D, 
observe also its direction from X, and determine both 
angles, then with the three elements tlms obtained (being- 
like to those of the second three in the preceding- 
process), we may determine the distance from B to X, 
and then, having the distances and directions from D to 
N, and from D to X (the figure being now considered 
as folded), and determining the included angle, we may 
by such elements (being like to those of the first three 
in the preceding process) determine the distance from 
~N to X. This is analogous to the regressive process. 

Surely, in all this wonderful accord there must be 
something more than mere coincidence. "The invisible 
things of God are clearly seen, being perceived through 
the things that are made." 

§ 15. But the subject concerning which we set out to 
make investigation may be the summum genus instead 
of the infima species or individual, as hitherto, and in 
such case we shall find that the processes of both inves- 
tigation and reasoning will be in the exactly opposite 
direction, and that the maximus term, instead of the 
magnus, as hitherto, will be the subject of the ultima, 
and the magnus term instead of the maximus will become 
the predicate. 

Strictly speaking, the word ' ' predicate ' ' is not prop- 
erly applicable to the last, but rajfcher to the first term of 
propositions as they will be exhibited in this section, 
inasmuch as the species cannot be predicated of the 
genus, but the genus of the species. To change the 



SOEITES. 



65 



names of the terms as they stand related to the proposi- 
tions would, however, be confusing, and they will, there- 
fore, be retained in their grammatical rather than in 
their strict, logical signification. 

But we shall find it necessary to change the signifi- 
cation of the copula. As hitherto employed, such sig- 
nification has been "is" or "is not" in the sense of "is 
(or is not) comprehended in" but as employed in this 
section only, the copula must be understood to signify 
" comprehends " ot " does not comprehend.'''' The rea- 
son for this change, if not immediately obvious, will 
become clear as we progress. It will, however, be here- 
after seen that in some cases the two significations are 
interchangeable, and either may be understood. 

I shall have immediate recourse to illustration by 
means of geometrical figures, as thereby such illustration 
can be made much clearer, being exhibited to the eye as 
well as to the understanding; and I now give, the fol- 
lowing figure, 




66 LOGIC AS A PURE SCIENCE. 

which you will observe is like to our original card- 
board figure on page 59, with triangles 1 and 3 remain- 
ing in the same position as therein, but with triangles 
2 and 4 turned upward, each in a semicircle, on the 
points D and J as centres respectively. 

The points XXX are now brought together in the 
figure, and jST 1ST N separated and become exterior. The 
points DDD and J J J retain their intermediate posi- 
tions. 

If now we begin to make investigation concerning X 
as the subject, we shall find ourselves proceeding in a 
descending instead of ascending direction, as before ; and 
we shall also find that the notions which we discover as 
predicabie (in the sense of the copula, as above changed), 
of our successive subjects, instead of being higher genera 
and comprehending the subjects, are lower species, and 
are wholly comprehended in the subjects respectively. 
The propositions in which we lay down our judgments 
will therefore necessarily be required to signify this dif- 
ference, which may be done by putting the predicates in 
capitals instead of small letters, as before, and will be as 
follows : 

X — J (meaning All X comiireTiends all J) ; 
J ■ — - D (meaning All J comprehends all D) ; 
D — 1ST (meaning All D compreliends all N). 

The subject of each of the foregoing propositions is 
distributed. But it might have been undistributed in 
so far as relates to the manner of its representation, and 
the proposition still retain its character as universal. 

To illustrate, I now reproduce the first combination of 
circles shown in the former part of this treatise when 



SOEITES. 



67 



treating of simple Syllogisms, adding another circle to 
make it applicable to a Sorites, the letters being pnt on 
the lines of the circles, and to be considered as indicating 
the whole areas included in the circles respectively. 




It will now be manifest, from mere inspection of the 
figure, that what we have predicated of X (viz., J) might 
also have been predicated of x, and in fact with more 
correctness, for J is comprehended loTiolly and only in 
that part of X which lies within the circle marked on 
one side J and on the other x. In like manner, what we 
have predicated of J (viz., D) might have been predi- 
cated of j, and what we have predicated of D (viz., N) 
might have been predicated of d. 

The propositions may therefore be stated as follows : 

X or x — J ; 
J or j — D ; 

D or d - N. 

In either alternative the propositions must be re- 
garded as universal. I shall hereafter make use only 



68 LOGIC AS A PURE SCIENCE. 

of that iii which the subjects are represented by small 
letters, as apparently, but not in fact, undistributed. In 
reading the propositions, the words "All " and " Some " 
must be expressed, and it must be borne in mind that 
the word "Some" applies to a definite part of the 
term, and when in the process of the reasoning a term 
with that word prefixed shall be repeated, it must be 
read or understood as "The same some," or "The 
same definite part of." 

The dictum of Aristotle, as applicable to the above 
propositions, will now have to be changed so as to read 
as follows. 

"Whatever definite term is affirmed or denied as com- 
prehending any other definite term, may be affirmed or 
denied as comprehending any definite term compre- 
hended in the definite term so comprehended, and in 
like manner of any definite term comprehended in the 
definite term so secondly comprehended, and so on ad 
infinitum. 

Applying the dictum as thus changed to the above 
23ropositions, the two forms of the full Sorites warranted 
therebv will be as follows : 



In the regressive 


In the progressive 


configuration. 


configuration. 


a — x. 


x — J, 


i - v, 


J -D, 


x- J; 


d- X; 


\ x — X. 


.-. x - X, 



and the abridged form will be 

... j _ D ; .-. x — X. 



SORITES. 69 

All propositions put forth in the above form in the 
descending processes of investigation and reasoning, 
may be converted simply, provided the original signifi- 
cation of the copula be at the same time reinstated, and 
by simple conversion of the above, we shall have the two 
forms of Sorites as we have hereinbefore seen them. 

But not only have the terms of all the propositions in 
the two forms changed places, but also the forms them- 
selves, in respect to the configurations, the converse of 
that which before was regressive having become progress- 
ive and of that which was progressive, regressive. By 
examining our original card-board figure in connection 
with the figures on page 62, and the remarks on the lat- 
ter, and comparing them with the first figure in this sec- 
tion, and applying such remarks to the configurations as 
herein given, it will be seen that such change is proper, 
triangles 3 and 4 in the latter figure being the analogue 
of the Sorites in the regressive configuration, and 1 and 
2 of that in the progressive. 

In like manner, it will be found that in all matters of 
form there will be continued inversions. 

The Sorites herein given may be expanded into series 
of Syllogisms as follows : 

In the regressive process. 
j-D, x - J, 

d-N; ■ j-N; 

.-. j — N.— — -^^ .-. x — N. 

In ilie progressive process. 
X J, ^- X 

j - D ; d 

.-. x — D 




70 



LOGIC AS A PUEE SCIENCE. 



All the propositions in the foregoing forms are uni- 
versal, but they may all be particular in the manner of 
their representation (indicated by the apparent non-distri- 
bution of the predicate), provided the definiteness of the 
terms represented be kept in view. Thus, in the fol- 
lowing figure, let the letters on the lines of the circles 
refer to the whole areas of the circles respectively as 
before, and those in areas only to the areas as bounded 
by lines respectively, but considering them where occur- 
ring more than once as to be taken together : 




The Sorites exemplified will be as follows : 



hi the regressive 


In 


the progressive 


configuration. 




configuration. 


d — 11, 




x — h 


.1 — cl, 




j — a, 


x — j ; 




( d — n ; 


.-. X 11. 




.-. x — n : 



and may be expanded into series of Syllogisms, as fol- 
lows : 



SORITES. 71 

In the regressive p>rocess. 

3 — , d > x — h 

d — n; ^ j — n; 

.-. j — n. —^ .: x — n. 

7?i tfAe progressive process. 

x — j, ^ x — d, 

j — d; t d — n; 

.-.x — d. — .-. x — n. 

Here apparently we have two anomalies — viz., Syllo- 
gisms having the middle terms undistributed in both 
premises, and Syllogisms in which conclusions are de- 
duced from particular premises. But they are such 
only in appearance ; all the propositions (the definiteness 
of the terms being kept in mind) being in fact universal, 
and the middle term distributed in each case in the 
major premise. 

The terms of all the foregoing propositions may 
each be considered as comprising all the areas marked 
in the figure with the small letters representing them 
respectively, taken together respectively, or only those 
areas respectively, in which the letters representing 
both the subject and predicate appear, taken together. 
In the former case, the subjects will each be greater 
than their predicates respectively, and the copula must 
signify "comprehends" in the latter, the terms of 
each proposition will be co-extensive, and the copula 
may have either signification. But in the latter case, the 
major middle term in the middle premise will narrow in 
signification to that of the minor-middle term, and the 
maxim us term in the ultima will have a narrower sig- 
nification than as employed in the maximus premise. 



72 LOGIC AS A PURE SCIENCE. 

The middle premise, it will be seen, has become the 
major premise, and the magnus premise the minor of the 
first Syllogism of the series in the regressive process, 
and the middle premise has become the minor, and the 
maximus the major of the first Syllogism of the series in 
the progressive. The regressive Sorites in the descending 
process is therefore, in the forms above given, which you 
will find are the perfect forms, a major Sorites, instead of 
a minor, and the progressive a minor Sorites instead of a 
major as before. It will be also seen that the Enthymeme 
taken from the second Syllogism in the regressive series 
is of the second instead of the first order as before, and 
vice versa in the progressive. If a synopsis should be 
made, this would necessitate (to make it conform to the 
former) the transfer of the headings of the columns on 
each side of each page of the former from each page to 
the other, and their transposition after being so trans- 
ferred. 

All the Syllogisms are in the fourth figure, which in 
this process becomes the perfect figure, the first be- 
coming imperfect. The second and third figures will 
also be found to have changed places, if indeed they 
and the first can have any place at all, in the new 
sense of the copula. One of the premises in each case 
in the second and third figures, and both in the first, 
would necessarily be in the inverse order, affirming or 
denying of the species that it comprehends or does not 
comprehend the genus, or else the original signification 
of the copula would have to be considered as reinstated 
in such premises, and the process would thereby lose 
its distinctive character as a process wholly in the 



SOKITES. 73 

descending direction, which only we are now considering. 
By examining the synopsis, it will be found that in all 
cases in which either of the involved Syllogisms in the 
columns on the left side of the regressives or right side 
of the progressives is in one of the imperfect figures, 
and in all cases of combinations of Syllogisms shown 
on the other side of each page respectively, the process 
of the reasoning partakes of both characters, being partly 
in the ascending and partly in the descending direction. 
I shall not proceed further with the consideration 
of this subject, for the reason that propositions in the 
descending process are seldom, if ever, put forth in form 
as herein given, but in the converse. When you come to 
the study of Logic as illustrated by concrete examples 
(in which aspect it is, in respect to each such illustration, 
an applied science), you will find a distinction made in 
respect to the quantity of concepts (terms) as being either 
in extension or intension, the latter being called also 
comprehension. This distinction runs also into the prop- 
ositions and syllogisms as treated of, according as the 
terms are considered as in one or the other quantity. 
You will find it, however, to be of no practical impor- 
tance in so far as the process of reasoning is concerned, 
all reasoning being conducted on the lines of the pro- 
cess, as we have previously considered it, and being 
called reasoning in extension, in contradistinction to the 
process as shown in this section, which is called reason- 
ing in intension or comprehension. The distinction, in 
so far as it relates to the terms (concepts), does not lie 
within the province of Logic as a Pure Science, and 
cannot be illustrated by means of symbols indefinite 



74 LOGIC AS A PUEE SCIENCE. 

in material signification, but the illustration of the pro- 
cesses of investigation and reasoning wholly in the 
descending direction, given in this section, will serve to 
make it, as continued into the reasoning process, clearer 
and more easily understood. 

The consideration of the subject matter of this sec- 
tion would perhaps have been more appropriately intro- 
duced when treating of simple Syllogisms, but it could 
not have been made as intelligible without as with geo- 
metrical illustration by combinations of triangles, and 
the latter has been more appropriately, and at the 
same time more effectively, introduced in this chapter, 
where it has been exhibited in one view, and to its full 
extent. 

The copula must now be considered as returned to 
its original signification, and where the word ' ' descend- 
ing" shall be hereinafter used, it must be considered 
as applicable to the direction of the process of inves- 
tigation, but not to the form of the propositions, which, 
in the perfect moods of the Sorites, will always be found 
in the converse of those herein given. 

§ 16. Thus far the premises of the Sorites exhibited 
have consisted of propositions put forth independently 
as the results of investigation. They may, however, be 
the results of prior processes of reasoning, the premises 
of which may be required to be exhibited in connection 
with them, in order to a clear understanding of the prin- 
cipal argument. The full expression in such case will 
become complex, and may be in two forms, of which I 
first exhibit the following : 



SORITES. 75 

J X, V Z — s and Y — z and J — y. 

D — j, v B — j and D — b. 
1ST — d, V K — d and N — k ; 

" .-. N— x. 

Here each premise is the ultima or conclusion of a 
prior process of reasoning, the premises of which are 
affixed, with, the word "because" preceding. 

In the example, all the premises have supporting 
premises affixed. But any one, or two, only, may have 
such premises affixed, the other two, or one, as the case 
may be, being propositions put forth independently. 

The whole expression, in either case, is called an 
Epicheirema, or Reason-rendering Syllogism (of either 
three or four terms). The principal argument, with ref- 
erence to the supporting premises, is called an Episyllo- 
gism ; and the supporting premises in each case, with 
reference to the premise proved, is called a Prosy]logism. 

The second form is that in which the premises of the 
Prosyllogism are prefixed, those in relation to the first 
premise being stated antecedently to the whole principal 
expression ; those in relation to the second or middle 
premise, interpolated between the first and middle, and 
those in relation to the last, interpolated between the 
middle and last. 

If either of the first two be in such form, it will be 
found upon trial, that the principal expression has lost 
in forcibleness of statement or in perspicuity, and they 
may, therefore, be disregarded, but the third will be 
found to lead to greater perspicuity, and especially so if 
more than two new middle terms are called into requi- 
sition for the purpose of elucidation. 



76 LOGIC AS A PUEE SCIENCE. 

The first form (Epicheirema) is better adapted to the 
statement of arguments in which the premises are ex- 
plained, the second to those in which either the first or 
last premise is disputed. It is seldom the case in anj^ 
disputation, that more than one of the premises of the 
principal argument is called in question, and that one 
is generally the first or last, the middle premise being 
usually a general rule acquiesced in upon being stated ; 
and if the disputed yjremise be the first, the principal 
argument, by changing the configuration, may be thrown 
into such form that it shall become the last. 

I now proceed to consider Sorites as complex expres- 
sions, in the second form, but only as limited to those in 
which the last itemise is disputed, and to distinguish 
them as such, shall call them Compound Sorites. 

§ 17. A Compound Sorites, once compounded, when 
fully expressed, consists of a simple Sorites (herein 
called the principal Sorites) with two, or three, proposi- 
tions interpolated between its middle and last premises ; , 
such propositions (if there be two) constituting the pre- 
mises of a simple Syllogism of which the conclusion, or 
(if there be three) of a simple Sorites, of which the 
ultima is the last premise of the principal Sorites. The 
interpolated propositions will be herein called the in- 
cluded Enthymeme, if there be two, or Sorites, if 
there be three, giving the fall name in the latter 
case, in default of one analogous to Enthymeme in the 
former. An included Sorites may in like manner 
have an Enthymeme or second Sorites included within 
it, and the second included Sorites may in like manner 
have an Enthymeme or third Sorites included within it, 



SORITES. 77 

and so on ad infinitum. There can be bnt one included 
Enthymeme, and it will always be the last included ex- 
pression. The reasoning in all such cases, while it will 
have the ajDpearance of being very much involved, will 
in reality be very much clearer. 

§ 18. But compound Sorites are seldom, if ever, fully 
expressed in formal, prepared argumentation, the last 
premise of the principal Sorites being suppressed, but, 
as will be hereinafter shown, in all cases implied. In 
this aspect a compound Sorites may be better denned 
as an argument consisting of more than four expressed 
propositions composed of as many terms as there are 
expressed propositions, including the ultima. Both 
definitions will be better understood by illustration. 

Let us suppose the case of two disputants of whom 
one, the proponent, advances these propositions : 

D - U 

.-. N — x, 

to which the other, the opponent, answers : I admit that 
D — j, but I deny that it follows that 1ST — x. 

The propositions, as you will observe, constitute the 
abridged form of the first mood. 

The proponent replies, asserting, as the reason, the 
two propositions necessary to make up the expanded 
form, viz.: 

•/J — x 
ancb-JNT — d, 

and to this the opponent makes rejoinder: Admitting 
that J — x, I deny that N — d. 



78 LOGIC AS A PUEE SCIENCE. 

The issue is now clearly defined, and the whole case 
may be stated as follows : 

J — x admitted, 

D — j admitted, 

1ST — d alleged but denied, 

.". IS' — x claimed but denied. 

The proponent, to maintain the issue on his part, 
must establish that IS 1 " — cl, or must fail. 

To do it, as the proposition to be established is A, he 
must find a middle term, with which both the terms 1ST 
and D may be compared, so as to form, with the con- 
clusion, a perfect Syllogism in Barbara (symbols AAA), 
or two middle terms, with one of which 1ST may be com- 
pared and D with the other, and one of which may be 
predicated of the other, all in such manner as to con- 
stitute, with the ultima, a valid Sorites in the first mood 
(symbols A AAA). 

Let the middle term, in the first case, be Y, and the 
two middle terms, in the second case, be Y and Z. 

The Syllogism in the first case will be : 

Y — d, 

N - y; 
.-. N — d. 

But in the second case the two new terms are required 
to he compared, and either may be the subject of the 
proposition in which they are compared, viz., Y — z or 
Z — y. The abridged Sorites may therefore be either : 

Y — z ; .-. N — d ; 

or, Z — y; .-. 1ST — d. 



SORITES. ' 79 

Let ns take the first, and in order to expand it into 
a full Sorites, let us write down the first mood in the 
regressive configuration, as in the synopsis, and write 
under its second and fourth propositions the abridged 
Sorites thus taken, as follows : 

J — x; D — j; N — d; .-. N — x. 
Y — z; ..-. N — d. 

Then, by expressing the first and third implied 
propositions of the abridged Sorites (making them to 
correspond in respect to the terms employed), we shall 
have the expanded Sorites as follows : 

* Z — d; Y — z ; N — y; .-. N — d. 

By taking from the Syllogism in the first case its two 
premises (constituting an Enthymeme of the third order), 
and from the Sorites in the second case, its three pre- 
mises, and interpolating them (respectively)- between the 
middle and last premises of the principal Sorites, we 
shall have, in each case, a compound Sorites fully ex- 
pressed, as follows : 



In the first case. 


In 


the second case 


J - x, 




J — X, 


D- h 




d - i, 


Y-d, 




Z — d. 


N -y; 




Y- z, 


.-. n- 




N — y; 


N — d, 


.' 





and .*. N — x. 




N — d, 




and .• 


. N — x. 



80 LOGIC AS A PUEE SCIENCE. 

The conclusion of the first Enthymeme of the princi- 
pal Sorites, viz. , D — x, is held in the mind ready to 
unite with the last premise, 1ST — d (after the latter shall 
have been proved), in establishing the ultima, 1ST — x. 

§ 19. But there is a shorter and simpler process, and 
the one which is usually employed informal, prepared 
argumentation. Instead of holding in the mind the con- 
clusion of the first Enthymeme to unite with the last 
premise of the principal Sorites when proved, as above 
stated, we may at once employ it (mentally) as a premise 
in connection with the first of the new expressed proposi- 
tions, and in like manner the unexpressed conclusion re- 
sulting from them as a premise in connection with the 
second new expressed proposition (and in the second case 
as above, the unexpressed conclusion thus resulting in 
connection with the third), and shall find that the last 
premise of the principal Sorites will not appear. Thus, 
in the two cases, the unexpressed conclusions being given 
in italics : 



In the first ease. 


In the second case. 


J -x, 


J — X, 


D .— ■ j, (.'. D-x). 


D — j, (.-. D - x). 


Y — d, (.-. y—x). 


Z — d, (.-. z — x). 


^ — y; 


Y Z, (.-. Y—x). 


■. N" — x. 


N.-y; 




.-. N — x. 



But the last premise of the principal Sorites will have 
been implied, as will be manifest from a comparison of 
the two forms in the second case put side by side, as 
follows : 



SOEITES. 




m in second case. 


/Second form in secc 


J - x, 


J — X, 


J) — i • (.-. D— x, held in the mind), 


. D - j, 


_ _ _ _ 


Z — d, 


Z — d, 


Y - z, 


Y — z. 


JST - y ; 


N- y; 


.-. N — x. 


N — d; 


- 


N — x. (••• D-x). 





81 



and 

The second form is the simpler, but the first is the 
clearer, exhibiting the entire process of the reasoning. 

The included Enthymeme in the first case, or Sorites 
in the second, serves only to prove the last premise of 
the principal Sorites, and forms no part of the argument, 
which is wholly contained in the principal Sorites. 

§ 20. Both the principal and included Sorites in the 
examples are in the regressive configuration, but they 
may be in different configurations. If in the foregoing 
disputation the opponent in his rejoinder had admitted 
the magnus premise, ~N — d, but denied the maximus, 
J — x, the principal Sorites of the proponent would 
have been in the progressive configuration, and the in- 
cluded one could have still been in the regressive, viz. : 





N — d, 




D- j; 




Z - x, 


v. 


Y - z, 




J — y; 




J _x, 


and . 


•. N — x. 



82 LOGIC AS A PURE SCIENCE. 

The two configurations cannot be directly linked! 
together in this example, as before shown in the second 
form, there being a break in the chain between the 
second and third propositions. Bnt by considering the 
configuration of the included Sorites to be changed (as 
it may be by transposing the first and third premises 
thereof), the whole expression can be put in the second 
form as before, and the last premise of the principal 
Sorites, J — x, will not appear. It does not, however, 
follow that the two configurations cannot in any case be 
directly linked together. That they may be in some 
cases will be hereinafter seen. 

§ 21. All the Syllogisms involved in all the foregoing 
examples are in Barbara, and the dictum of Aristotle, 
as hereinbefore extended, may be directly applied to 
those in the second form, by extending it still further in 
like manner. But to those in the first form it would 
have to be twice applied, first to the included Sorites 
and secondly to the principal, and in that case would 
not require to be further extended, both the Sorites 
being simple. 

§ 22. But if any of the involved Syllogisms are in 
any other figure, or combination of figures, they would 
have to be converted into Syllogisms in the first figure, 
before the dictum could be directly applied. 

The following are examples of compound Sorites, 
the involved Syllogisms of which are in combinations of 
figures, as shown by the names of the moods given in 
connection with them. The conclusions proved, but not 
expressed, are also given in italics in connection with 
the names of the moods, except the ultima of the in- 



SORITES. 83 

eluded Sorites (in the first forms), which is expressed as 
a premise below the second dotted line. The principal 
Sorites and the number of its mood and the configuration 
are given in advance of each example : __ 





Gill Regressive Mood. 




J 


— x, d — j, D -.— n; .: n — 


- X. 


7 irst form. 


Second form 




J-x, 


J-x, 




cI — j; 


(.•. d — x, Darii), d - — J, ( 


.•. d—x, Darii), 





D — z, ( 


.'. z — x, Disamis) 


D— z, 


z-y, ( 


.-. y — x, Disamis) 


z-y, 


Y— n; 




Y — n; 


.*. n — x. 


(Disamis). 



D — n; 

and .'. 11 — X. (•.• d — x, Disamis). 



15th Progressive Mood. 
J _ N, D — j, X — d ; .-. N — X. 





First form. 




Second form. 




J— N, 




J — N, 






D-5; (■• 


^r — 2?, Camenes), 


D-J, 


(.•. &—D, Camenes), 









T-d, 


(.'. Sf — Y, Camestres), 




Y— d, 




Z-y, 


(.: If—Z, Camestres"). 




Z-y, 




X-z; 






X — z ; 


^- 


.-. N — X. 


(Camestres). 




X — d; 








and 


\ if — X. c. 


• 2?—D, Camestres). 







84 



LOGIC AS A PUEE SCIENCE. 







25th Regressive Mood. 


a- 


^X, 


D 


- h J 


— n ; .'. n -_ X. 


First form. 








Second form. 


d-^X, 








a-^x, 


d-d; 


(..-.j- 


-x, 


Bolcardo), 


D — j, (.-. j-~-X, Bokardo), 











J — J, (.-. y— X, Bokardo) 


J-y, 








Y — Z, (.-. z -~X, Bokardo) 


Y-z, 








Z — n; 


Z— n; 








.*. n --_> X. {Bokardo). 


J — n ; 










d .-. n-~X. 


C-i- 


-X, 


Bokardo). 





§ 23. The included Sorites may have an Enthymeme 
or a second Sorites included within it, and the second 
included Sorites may have an Enthymeme or third Sorites 
included within it, and so on ad infinitum. Thus : 



First form. 
J-x, 

D j ; (.-. D — x, held in the mind), 

j Y-d, 

( Z — y; (.-. Z — d, held in the mind), 



K— q, 

>- Second included Sorites, 

Q — n; 



Second form. 
J — x, 
D — j, (.-. D-flj), 

Y — d, (.-. r— a?), 
Z — y, (.-. z —x), 
z — k, (.-. k —x), 
K — q, (.-. ff — «), 
Q — n; 
.-. n — x. 



'. '.:. (••• z—d), 

d — n ; 
and .•. n — x. (v.z>— x). 






SOEITES. 85 

If the first included Sorites in the last example be 
put in the regressive configuration, its last premise will 
be Y — d instead of z — n, and the second included 
Sorites will be employed to establish the former instead 
of the latter, but of course by different premises. In 
such case we shall find that when we attempt to put the 
whole expression in the second form, the premises of the 
second included Sorites will take precedence of those in 
the first, and the latter will be transposed. Thus: 

First form. Second form. 







J X, 




J — X, 






D-j; 


(.". B — x, held in (he mind), 


D-3> 


(.-. B — x), 









Q-d, 


(.-. Q-x), 


A 


z — n, 




K-q, 


(.-. K-x), 






Z-y; 


(.'". y — w, held in the mind), 


Y-k, 


{.: Y-x), 











Z-y, 


(.: Z-x), 






Q-d,] 




i • — n ; 




« 


,J 


K-q, 
Y-k; 

• .. 
Y-d; 


► 


.'. 11 X. 




(. 


. ( 


• y — n), 








d — n ; 








a 


Qd . 


. u — X. 


(v B - x). 







The argumentation is supposed, of course, to have 
taken place on the lines of the process in the first form, 
and the second included Sorites did not therefore come 
into the process until the proposition, Y — d, was dis- 
puted. The illustration thus shows the superiority of 
the first over the second form, as exhibiting the whole 



86 LOGIC AS A PUKE SCIENCE. 

process of the reasoning. The second could not have 
been framed until the lirst had been gone through with. 

§ 24. Compound Sorites may, however, be exhibited 
in forms which at first sight may seem to be in contra- 
vention of what has been before laid down, but upon 
examination it will be found that such is not the case. 

Thus, in the two following cases : * 

(l.) (2.) 

N — d, N — d, 

D - j, D - j, 

J — X, J — x, 

Y — x, ¥ — X, 

Z - y, Z - y, 

Q — z; Q — z; 

.-. N — Q. .-.. N — Q. 

Let us take the second and write in line with each 
premise (except the first and last) the implied con- 
clusions : 

N — d, 

D — j, (.-. N-j), 
J — X, (.-. N—x), 

¥ — X, (.-. &- Y), 
Z — y, (.-. &-Z), 
Q — z; 
.-. N — Q. 

The expression, with the exception of the ultima, will 
be found, upon examination, to constitute the premises 
of two simple Sorites, of which the first is in the pro- 
gressive configuration and the second in the regressive. 

* Taken from Schuyler's Logic, p. 88. 



SOEITES. 87 

By stating them successively with their implied 
ultimas, we shall have them in the folio wins: form : 





n — a, 






D- J, 






J - x; 


(.•. iV — a;, held in the mind) 




¥ — X, 


* 




z - y, 






Q- z; 


(.". -^ — X Ae&Z in the mind) 


and then, 






:• N — x, 






•and -Q- — X\ 


.-. -§- ff; 




or, 






v -Q- - X, 






and N — x; ' 


;••, n - q. 





I now proceed to show that Sorites, stated as above, 
fall within the definition of compound Sorites, as herein- 
before given. 

The maximus premise, being the last premise of the 
principal Sorites involved in the foregoing examples, 
has not appeared, but has in all cases been implied. 
The middle premise is (as has been before stated to be 
the case in all Sorites, simple and compound) the second, 
and in the examples is D — \. Combining this with the 
ultima, the abridged form of the principal Sorites is 
therefore, 

d- i; 

.-. *F - Q. 



b8 LOGIC AS A PUEE SCIENCE. 

Expanding this in the 12th progressive mood as in 
the synopsis, we shall have the full principal Sorites as 
follows : 

N — d, D — j, J — Q; .-. ^ — Q; 

and the compound Sorites will be as follows : 

N — d, 

i) — J ; (.-. N—j, held in the mind), 





J 


J - X, 






[I 


¥ — X; 


(.'. <f — Y, held in the mind), 


■< 




z- y r 

Q — z; 


\ 




<i 


Q — y; 




(. 




(••• f- T), 




J - Q; 




ai 


id •■ 


. N— Q. 


(•■• If -J). 



§ 25. But the magnus and maximus terms of the 
principal Sorites, at the ultima of which we first arrive, 
may not be the infima species and summum genus, and 
further investigation may bring into the process of the 
reasoning lower species or higher genera, and if in both 
directions, both ; and the new term or terms, instead of 
being employed interiorly as middle terms as hitherto, 
will be employed exteriorly. In such case the new 
term, or terms, will constitute, if there be but one, 
a new magnus, or maximus term ; or if there be two, 
obtained by investigation in both directions, both, and 



SOEITES. 89' 

the displaced terms will become middle terms. We shall 
then find that there will be two new abridged and full 
principal Sorites in each case, one regressive and one 
progressive, but varying according as the new term, or 
terms, are applied to the original Sorites considered as 
both regressive and progressive. They will, however, be 
independent of each other, and each .will have its correla- 
tive in their respectively opposite configurations. The 
displaced original term, if it shall have been the magnus, 
will become the minor-middle term of the new principal 
progressive Sorites, and will not appear in the new re- 
gressive, but if the magnus term be again displaced by 
bringing in another, then the displaced original term will 
become the major-middle ; but if the displaced original 
term shall have been the maximus, then it will become the 
major-middle term of the new principal regressive Sorites, 
and will not appear in the new progressive, and if the 
maximus term be again displaced by bringing in another, 
the displaced original will become the minor-middle term. 

But of the original premises in the case of one new 
term being brought in, one, or two, will still remain in 
each new principal Sorites, one in the regressive configu- 
ration, and two in the progressive, if the new term Ibe 
maximus, and vice versa, if magnus. One original premise 
only will remain in each of the new principal Sorites in 
any case if two new terms, one magnus and one max- 
imus, are brought in. 

The original ultima-will of course have disappeared in 
every case. But if two new terms are brought in, both 
having been discovered in a process of investigation in 
one direction only, the original ultima will reappear as a 



'90 LOGIC AS A PURE SCIENCE. 

premise of one of the new principal Sorites, the regres- 
sive, if the investigation were in the ascending direction, 
and the progressive, if in the descending. 

If the investigation shall be pursued so that more than 
two new terms shall he brought in, in each direction, 
every vestige of the original principal Sorites will have 
disappeared from the new principals, as they will then 
be constituted. 

But all the premises of the original principal Sorites 
will, in all cases, be found to remain, either partly in the 
principal Sorites, and partly in the following included 
Enthymeme or Sorites, or in two of the included Sorites, 
or wholly in the last included Sorites, or partly in the 
Enthymeme, which is the final expression, and partly in 
the next preceding included Sorites, according as the 
new terms shall be brought in ; and they will always be 
found together in their original order, either regressive 
or progressive, how far soever the process be continued, 
and this, also, whether the compound Sorites be in the 
first or second form, as hereinbefore shown. 

The following examples illustrate all the foregoing 
remarks, except the last, as to compound Sorites in the 
second form, which can be verified by trial. All the in- 
volved Syllogisms are in the first figure throughout. 

The original premises and ultima (employed as a 
premise) are printed in Roman letters, and those which 
remain in the principal Sorites in full-faced type. All 
other propositions are printed in Italics. The examples 
having the same number of new terms are so arranged, 
either on the same or opposite pages, that they may be 
readily compared. 



SOEITES. 91 

With one new term, brought in in the ascending process of in- 
vestigation, and therefore a new maximus term: 



Regressive Configuration. 




Progressive Configuration. 


X - y, 






N — d, 


J — X ; (••• J- 


-y), 




D — j ; (.-. n- 3 \ 


D - j, 






J — X, 


N — d; 






x — y\ 


N - j; 






J — y; 


and .-. N — ■ y. (•.• j- 


y). 


and . 


. N — y. (•.• n-3). 



Full forms of neiv principal Sorites : 

X—y, J — x, N— j; /. N— y. 
IS" — d, D — j, J — y; :. N — y. 

With one neiu term, brought in in the descending process, and 
therefore a new magnus term: 



Regressive Configure 


ition 




Prog 


ressive Configuration. 


J — X, 








K — n, 


D - j; ( 


: D- 


-cc), 




N — d; (.-. k-cd 


N— d, 








D - j, 


K — n; 








J - x; 


K— d; 








I) — x; 


and .*. K — x. (•. 


D — 


X). 


and 


.'. K — X. (•.• K- d) 



Full forms of neiv principal Sorites : 

J — x, D — j, K — d ; .-. K — x. 

K — n, N — d, D — x; .: K — x. 



92 LOGIC AS A PURE SCIENCE. 



With two new terms, one brought in in the ascending process of 
investigation, and therefore a new maximus term, and the 
other brought in in the descending process, and therefore 
a new magnus term: 

Regressive Configuration. Progressive Configuration. 

X — y, K — n, 

J — x; (.-. j—y), N — d; {.-.k— <d, 

D - J, D - j, 

N — d, J — x, 

K — n; X — y ; 

% — j 5 D — y; 

and .*. K — y. (•.• -7— y). and ,\ K — y. v.-K—O). 

Full forms of new principal Sorites : 

X—y, J— x, K—j\ :. K — y. 
K — n, N — d, D — y ; .-. K — y. 



SORITES. 



93 



With two new terms, both brought in in the ascending process 
of investigation, and one therefore a neiv maximus term : 



■Regressive Configure 


ition 




Progr 


essive Configuration. 


Y — z, 








N — d, 


X — y;{.- 


X- 


-»>, 




D — j; (.-. iV-i), 


J — x, 








J - x, 


D - j, 








* x — y, 


N — d; 








Y- z; 


^s T — x; 








J - z; 


and .-. N — z. (-. 


■ X- 


■Z). 


and 


'. X — Z, (V N—J). 



Full forms of new principal Sorites : 

Y—z, X—y, N— x; .*. X—z. 
^— d, D — j, J—z; .-. iV 7 — z. 



FFtYA £wo we.?0 terms, both brought in in the descending process, 
and one therefore a new magnus term: 



Regressive Configuration. 




Progressive Configuration. 


J — X, 




q — h 


D - j; (•■• d- 


so), 


K — n; (.-. Q-rt) 


N — d, 




N — d, 


K — n, 




D - J, 


q — h; 




J - x; 


q - d; 




ff — x; 


and .*. q — x. (.• b — 


X). 


and .-. q — x. (v q — n) 



Full forms of new principal Sorites : 

J — x, D — j, Q — d ; .-. § — x. 

() — h, K — n, IV — x ; .-. q — x. 



94 



LOGIC AS A PURE SCIENCE. 



With three new terms, of ivhich ttvo are brought in in the ascend- 
ing process of investigation, and one of them therefore a new 
maximus term, and the third in the descending process, 
and therefore a new magnus term: 

Regressive, Configuration. Progressive Configuration. 

Y — z, K — n, 

X — y; (.-. x— 2), jST — cl; (.-. k— a), 



J - x, 

D — j ; (.-. D-x), 



n -a,] 

K — n; 



[i K-d; 

\.°. (v D — x), 

K — x; 
and .'. K — z. (■.• x— z). 



S D - j, 

[I J — x; (.-. d — x). 

Y — z:V 
I 
,. J 

{.: (•.• d-x), 

D — z; 
and .*. K — z. (•.• k-o). 



Full forms of new principal Sorites : 

Y—z, X—y, K—x; .: K — z. 
K— n, N — €l, D — z; .-. K—z. 



SORITES. 



95 



With three new terms, of which two are drought in in the descend- 
ing process of investigation, and one of them therefore a new 
magnus term, and the third in the ascending process, and^ 
therefore a new maximus term: 



Regres 


sive Configuration. Progressive Configuration. 




X-y, 




Q - h, 




J — x; (.-. J-y\ 




K — n ; (.-. q- n \ 


f 


D - j, 


J 


N — d, 




i 


& — d; (.\N-j), 


\ 


D — j; (.-. iV-A 


■> 




K-n,) 

Q -h;\ ■ 

i 

J 

Q —n; 




7_ x ;i 

x-y;Y 




.{ 


-{, 


J — 'y\ . 


(..• 


{:■ N-j), 


. (:■■ &-j), 




Q -j; 




N-y; 


a 


nd / 


Q — y. (•.• j-y). a 


nd .• 


Q — y. (v q-n). 



Full forms of neio principal Sorites : 

X—y, J — x, Q — j; .: Q — y. 

Q — Jc, K— n, N—y; .: Q — y. 



96 



LOGIC AS A PUKE SCIENCE. 



With four new terms, of which two are brought in in the ascend- 
ing process, and two in the descending : 



Regressive Configuration. 
Y — z, 
X — y ; (.-. x- z), 



Progressive Configuration. 

Q - h, 

E — n; (.-. Q - n), 





J - x, 

d - J; 

N - d/ 


(•• 


D- 


-x), 


-{ 


N — d, 

D - j ; (• 
J — i,] 


. N-J), 


< 


K — n, 


>- 




< 




X-y,\ 






Q -h; 








Y — z;\ 






. . 












'{'. 


Q — A\ 
Q — x; 


■ B- 


-x), 


4 


J — z; 

X — z; 


■■ N- J), 


and . 


. Q - z. 


(■ 


X- 


-z). a 


nd . 


: Q — z. < 


.' Q — n)o 



Full forms of new principal Sorites : 

Y- z, X—y, Q — %; ••• — 2- 

Q — lc, K — n, N— z; .: Q — z. 



SORITES. 



97 



With eight new terms, of which four are brought in in the 
ascending process, and four in the descending: 



Regress ive Configuration. 

s — t, 

Z — s ; (.-. z — t), 



Progressive Configuration. 

S- g, 

G — q; (.-. H-q\ 



,f Y - z, 

( X — y ; (.-. x— s), 



\ D 



y, (.-. b— x\ 



f N - d, 

1 K — n ; (.-. k- d), 



Q — h~ 
G — q, 
H-g; 



[S H-h; 

1 .'. (••• K-d), 



f II— d; 



(••■ D — x\ 



f H — x ; 

\ ■ " 



(■•• X- 2), 

and .'.if — /. (•.• z — f). 



j Q - *, 

I K — n ; (.-. q — w), 

J n — a, 

*• D — j; <•'■■ #"-/>. 

J J — X, 

f l X — y; {.:J-y), 



Y 


— 


z, ' 


z 


■^- 


s, 


s 


— 


t\ 


— . 


_ _ 


— 



f r-t; 

I .'. — (••• J — y), 

J — t; 

. (vX-j), 

( N — t; 

I .'. (••■ Q - n), 

Q - t; 
and .•.5" — t. (•.• fi-g). 



Full forms of neiv principal Sorites: 

S—t, Z—s, H—z; .'. H—t. 
H— g, G—q, Q — t ; .-. H — t. 



98 LOGIC AS A PURE SCIENCE. 

§ 26. To recur now to illustration by means of geo- 
metrical figures. 

A regular tetrahedron may by four sections, beginning 
in the middle of each of its edges and made parallel to 
the opposite planes respectively, be divided into five fig- 
ures, of which four will be regular tetrahedra, and the 
fifth and interior figure a regular octahedron. 

Thus, by reproducing our former illustration on card- 
board before folding, and dividing it by lines which 
shall represent the four sections, we shall have the 
following : 



Now, assuming each interior dotted line to be the 
edge of an equilateral triangular plane, represented by 
card-board, projecting backward, divergingly, at the 
proper dihedral angles, from the plane of the one which 
we are supposed to have in hand, then, by folding the 
latter as before, we shall have a combination of five 
figures, as above stated, which will present to our eyes 
successively, as we turn it about as before, the following 
figures : 



SOEITES. 

First Series. 



99 





Second Series. 





Each of the four tetrahedra having one original exte- 
rior point, and three visible and one invisible planes, will 
be found to have that point marked with one of the let- 
ters N, D, J, X on each visible plane ; the fifth figure, 
the octahedron, having no original exterior point, and 
four visible, and four invisible planes, will be found 
marked on each visible plane with one of the numbers 
1, 2, 3, 4. It is wholly included, and occupies all the 
space, between the invisible planes of the four tetra- 
hedra and planes connecting their visible planes, and 
its volume is exactly equal to the sum of their vol- 
umes ; and it may well be regarded as the analogue 
of the ultima conclusio of the Sorites, of which the 
abridged form is : 

D-j; .-. N-x. 

The analogy between a compound Sorites in which the 
original principal Sorites shall remain the principal, and 



100 LOGIC AS A PUKE SCIENCE. 

a Sorites be interpolated as hereinbefore shown, and 
a tetrahedron divided by sections as represented in the 
foregoing illustration, cannot be exhibited as simply or 
as clearly as that between a simple Sorites and a tetra- 
hedron considered as a unit, as in our former illustration, 
because the tetrahedron which is the analogue of the 
included Sorites is involved in and forms an indistin- 
guishable, but, as must be regarded, separate, part of 
the included octahedron, having one of the visible planes 
of the octahedron as its only visible face. Its invisible 
faces cannot be brought to the surface in the following 
figures, but must be regarded as represented by the three 
triangles by which its visible face is bounded, the ultimate 
point of which will be found marked X in. the figures. 
Its ultimate point will not be the point X as shown in 
the figures, but will lie in the perpendicular let fall from 
the point 1ST upon the opposite plane of the original 
tetrahedron. We shall hereinafter find that perpendic- 
ular to be part of one axis of a sphere produced by the 
revolution of the tetrahedron, and that the pole of that 
axis opposite N should be marked X. The ultimate 
point of the indistinguishable tetrahedron which is the 
analogue of the included Sorites, may be at any point 
in the line of this axis within the octahedron, and let us 
assume that point to be in the centre of the invisible 
plane of the octahedron opposite its visible plane which 
is the visible face of the involved tetrahedron. The in- 
visible faces of the latter will then be equal to the tri- 
angles by which its visible face is bounded in the figures. 
Let us suppose that in the progressive process we 
have established the relation between N" and J, as in the 
lower one of the following combination of triangles 



SOEITES. 101 

(which, observe, are the same as the triangles 1 and 3 
in our original card-board illustration), and that the 
relation between J and X requires to be established. 




We shall then have the upper triangle in which only the 
relation (length of line) between D and J is known, and 
let us suppose that the relation between each of those 
points and X is not capable of being immediately deter- 
mined, but that there are two points (middle terms), one 
in each of the other two lines, capable of being succes- 
sively reached from D or any point in the line D J ex- 
cept the point J, and the length of a straight line con- 
necting them capable of being measured, and from both 
of which the direction of X can be observed, and the 
angles therefore determined. 

Reproducing the upper triangle and marking the mid- 
dle point in the base line J', and the two points at the 
extremities of the bas« X' and X", and the two new points 
Y and Z at the middle of each of the two lines connect- 
ing the extremes of the base with X, and connecting such 
new points, and each of them with J', we shall have the 
following : 



102 



LOGIC AS A PURE SCIENCE. 




and we may now establish that J' — X in the same man- 
ner as we have hereinbefore established that 1ST — X. 

But, the lines X' X and X" X are, by construction, 
equal to J X and D X in the upper triangle, on the 
preceding page, the middle points in which may be 
marked Z and Y. In the process we have found J' Z 
equal to J' Y, and X' Z equal to J' Z. X' Z is therefore 
equal to J' Y. But X' Z is J Z. And as J' Y is equal 
to J' Z, it will, upon being applied to the latter, coincide 
with it, and the point Y will fall upon the point Z. 
J Z may therefore be called J Y, and is equal to J' Y. 

The whole combination of triangles will now be as fol- 
lows, the original letters being put on the outside : 

x 




SOEITES. 103 

We can now express the full compound Sorites, ex- 
emplified by the foregoing illustration, as follows : 

N — d, 

D — j ; (.•. JV —j, held in the mind), 

J ■ — ■ y, (= JZ = J' Y, from which latter direction of X observed), 
Y — Z, (=Y Z, relation, i. e., length of line knoivn), 
Z — X 5 (- Z X, direction observed from former), 

J — x ; 

and .-. 1ST — x. (■.•N-j). 

This is the same compound Sorites as that exhibited 
in § 20,- on page 81, but with the included Sorites in the 
progressive, instead of the regressive, configuration. 

But if the interpolated expression be an Enthymeme, 
the analogy will be much clearer, as the lines by which 
the Enthymeme will be represented will lie wholly in the 
surface and not involve any section of the original figure. 

Thus, if in the following combination of triangles 
(which observe are the same as triangles 3 and 4 in our 
original card -board illustration) : 




we shall, in like manner as before, have established the 
relation between N" and J (as in the upper, left-hand 
triangle), from which latter we can see X, but are 
unable immediately to determine its distance, with- 



104 



LOGIC AS A PURE SCIENCE. 



out the knowledge of which we cannot establish the 
relation between N and X ; we may select another me- 
diate point, Y, which can be reached, and distance 
measured from J, and from which X may also be seen, 
and the angles therefore determined, as in the following 
fieri] re : 




and then, by the elements thus obtained, we can deter- 
mine the required distance from J to X, and by means 
thereof and the elements previously obtained, the dis- 
tance from 1ST to X. 

The compound Sorites exemplified by the foregoing 
illustration will be as follows : 





N — d, 






D - j ; < 


.-. N — j heldin the mind), 




Y- x, 






J -y; 






J - X, 




and . 


•. M" — x. 


v N-j). 



But if, instead of having begun in the ascending 
direction, we shall have begun in the descending, and 
have established the relation between X and D, as in 
the lower, right-hand one of the following combination of 
triangles (1 and 2 in the figure on page 65) : 



SORITES. 



105. 




and shall then, although able to see N from D, but not 
from X, be unable to determine its distance frorii D, 
without the knowledge of which, it would be impossible 
to determine its distance from X ; we may, in like man- 
ner as before, select another mediate point K, which can 
be reached from D, and from which N can also be seen, 
as in the following figure : 




and then, as before, may determine the required distance 
from D to X, and by means thereof and the elements 
previously obtained, the distance from X to X. 

The compound Sorites exemplified by the foregoing 
illustration, will be as follows : 



X comprehends J, 
J comprehends D : 

D comprehends X, 
K comprehends X J 



(.'. X comprehends D, held in the mind), 



and 



D comprehends X J 

X comprehends X. (v X comprehends D). 



106 LOGIC AS A PURE SCIENCE. 

By putting together the first of each of the two sets 
of figures in the preceding illustrations, on the line D J, 
common to both, we shall have the following figure : 




which is the same as that on page 61, but in a different 
position. By turning triangle 2 downward in a semi- 
circle on the point D as a centre, we shall have our 
original card-board figure ; or by turning triangle 4 up- 
ward to the like extent on the point J as a centre, we 
shall have the figure shown on page 65. Triangles 1 and 2 
taken together and 3 and 4 taken together are analogues 
of progressive Sorites, 1 and 2, in the descending direction, 
and 3 and 4, in the ascending ; but if 2 and 4 be both 
turned as above described, they will become analogues of 
regressive Sorites in the respectively opposite directions. 

§ 27. All the four triangles in our original card -board 
illustration are equilateral and equal. The solid figure 
resulting from the folding of the card-board is a regular 
tetrahedron, which is defined as a solid having four 
faces, all equal equilateral triangles. But the triangles 



S0KITES. 107 

might have been all equal isosceles triangles, or partly 
equilateral and partly isosceles. Such can be exhibited 
in- a plane figure bounded, by three, or four exterior lines, 
if the triangles are all equal, or by six, if they are partly 
equilateral and partly isosceles, and capable of being 
folded so that the exterior points shall meet in a perfect, 
but not regular, figure. But a perfect tetrahedron may 
have all its faces unequal, and in such case the faces may 
be spread out in an irregular plane figure having five 
exterior lines. In all cases the number of exterior lines 
will be found to be six, if bisected lines are counted each 
as two. All other plane figures having all the points 
exterior are imperfect and cannot be folded, so that the 
exterior points will meet, and their areas, and conse- 
quently the volume of space which they can be made 
respectively to inclose, can only be determined by means 
of the triangle. Imperfect Syllogisms and Sorites in 
logic must be reduced to the perfect figure before they 
can be submitted to the dictum de omni et nullo. 

"% 28. On the other hand, a tetrahedron (regular or 
perfect) may be added to on the outside by superimpos- 
ing on each of its faces another tetrahedron having a 
similar face, so that there shall be five tetrahedra in 
all. Four new points will have been added, all exterior 
to the original figure, the original points becoming in- 
terior, but their locations visible, the original figure 
having otherwise wholly disappeared from view. 

Similarly, as we have before seen, in respect to a 
Sorites, when four new terms have been brought in ex- 
teriorly, two in each direction, the four propositions of 



10*8 LOGIC AS A PURE SCIENCE. 

the original principal Sorites will have disappeared from 

the two new principals, as they will then be constituted, 

but they will remain in the included Sorites, of which 

the inner tetrahedron is the analogue. 

But in the figure, formed as above described, the four 

new points, which we will consider as marked K, Q, Y, 

and Z, will furnish only one new principal Sorites, as its 

analogue, which may be rendered in its abridged form 

thus : 

v Q-y; .-. K-z. 

But observe, the interior figure in the foregoing com- 
bination is a tetrahedron, not necessarily regular, but per- 
fect ; and if, instead of beginning with such a one, con- 
sidered as a unit, we begin with a regular one considered 
as divided by four sections, as before shown, and super- 
pose upon each of the visible planes of the included 
octahedron, a tetrahedron similar to each of the four 
resulting from such sections, we shall have a solid figure 
in the form of an eight-pointed star, the octahedron 
having entirely disappeared from view, except that the 
locations of its points will be visible. This eight-pointed 
star will be found to consist of two equal intervolved regu- 
lar tetrahedra, to both of which the interior octahedron 
will be common, and its revolution about its centre will 
produce a sphere exactly equal to that produced by the 
revolution of the original tetrahedron. Four exterior 
points will have been added, but of these two are the 
opposite poles of the two original points marked N" and 
X, and, having a common relation with them to the in- 
cluded octahedron, should be marked X and K" respect- 
ively, leaving, in fact, but two new independent points, 



SORITES. 109 

which may be marked Y and Z. The whole figure will 
then be the analogue of two independent full Sorites, of 
which one only is new, and that only in part, the abridged 

forms being : 

v D - j ; .-. N - x. 

..• Y — z ; .\ N — x. 

By comparing the foregoing illustrations with the 
Sorites having four new terms added exteriorly, given 
on page 96, the superiority of the Sorites over its ana- 
logue, the tetrahedron, will again be manifest. 

§ 29. Thus everywhere, whether we go inwardly or 
outwardly, and in all things, metaphysical as well as 
physical, we find triniunity, and can thence proceed to 
quadriunity, but beyond that, except in composite 
forms, we cannot go. 

§ 30. From the foregoing definitions and illustrations 
of Sorites, simple and compound, it seems manifest that 
the human mind is limited to reasoning concerning the 
relations of four terms. If other terms are brought in, 
they must relate to the terms of the principal argument, 
and in such case, if such relation be to the middle terms, 
they serve only to elucidate, but if to the magnus and 
maximus terms, then they supplant those terms ; which, 
if there be one, or two successively of each (new magnus 
and maximus terms) respectively, become terms of the 
two new middle premises respectively, but if more 
than two of each, Hhen are relegated to the subordi- 
nate position of middle terms employed only in elucida- 
tion. Otherwise they must be the terms of independent 
arguments. 



110 LOGIC AS A PURE SCIENCE. 

§ 31. There remains but to say that I have not pointed 
out the characteristics of Sorites, nor given the rules in 
relation to them, as the same have been usually pointed 
out and given (or in part so) in logical treatises, and to 
which reference has been hereinbefore made ; and I now 
refer to them only for the purpose of showing their 
inadequacy. 

They have been written with reference to Sorites 
treated of as capable of being expanded only in Syllo- 
gisms wholly in the first figure, and without reference, of 
course, to the distinction between them as simple and 
compound, which has been hitherto unobserved. They 
relate, 

1st. To the number of Syllogisms involved, as equal 
to the number of middle terms, and as ascertainable from 
the number of premises of the Sorites, less one. 

2d. To the character of the premises of the involved 
Syllogisms, whether minor or major, and the number of 
each and their sequence, viz.: one only, and that the 
first, major, and all the following minor in a regressive 
Sorites ; and nice nersa, in a progressive. 

3d. To the number and positions of particular and 
negative premises in the two configurations, viz. : that 
one only can be particular, and that the last, and one 
only negative, and that the first, in a regressive Sorites ; 
and nice nersa (in respect to positions) in a progressive. 

The first is true of all Sorites, simple and compound, 
in respect to the number of Syllogisms involved being 
equal to the number of middle terms, and has been im- 
pliedly shown as true of all simple Sorites, in respect to 
such number being ascertainable from the number of 



SORITES. Ill 

premises less one, in that they have been described as 
having three premises, and as being capable of expan- 
sion into two Syllogisms ; but in such latter respect it 
does not apply to compound Sorites when fully ex- 
pressed. 

The second, by an examination of the synopsis, will 
be found to hold good, of all regressive simple Sorites in 
respect to the moods in which they are minors, and not 
good in respect to those in which they are majors, and 
vice versa of all progressives. 

The third is of course, and for obvious reasons, appli- 
cable to all simple Sorites (but not to all compound, 
when fully expressed), so far as the number of particu- 
lar and negative premises is concerned, but to state it in 
respect to their positions as applicable to all Sorites 
capable of being- expanded in Syllogisms wholly in the 
first figure, and also to some in combinations of figures, 
either partly or not at all of that figure, and then to 
point out the very numerous exceptions in other like 
cases, would tend rather to confuse than to enlighten \ 
and I therefore leave the subject, and pass on to the con- 
sideration of Fallacies. 



